2. Multitype GWI processes

Let $\mathbb{Z}_+$ , $\mathbb{N}$ , $\mathbb{R}$ , $\mathbb{R}_+$ , and $\mathbb{R}_{++}$ denote the set of non-negative integers, positive integers, real numbers, non-negative real numbers, and positive real numbers, respectively. For $x,y\in\mathbb{R}$ , the minimum and maximum of x and y are denoted by $x\wedge y$ and $x\vee y$ , respectively. For $k,m\in\mathbb{Z}_+$ with $k<m$ , we define $\binom{k}{m}\,:\!=\,0$ . For $i,j\in\mathbb{Z}_+$ , let $\delta_{i,j}\,:\!=\,0$ if $i\ne j$ , and let $\delta_{i,j}\,:\!=\,1$ if $i=j$ (known as the Kronecker delta). For functions $f\,:\,\mathbb{N}\to\mathbb{R}$ and $g\,:\,\mathbb{N}\to\mathbb{R}$ , the notation $f(k) = \operatorname{O}\!(g(k))$ , $k\in\mathbb{N}$ , means that there exists a positive real number M such that $\vert f(k)\vert\leq M g(k)$ for each $k\in\mathbb{N}$ . More generally, for $d\in\mathbb{Z}_+$ and functions $G\,:\,\mathbb{R}^{d+1}\to\mathbb{R}$ , $g_1,\dots,g_d\,:\,\mathbb{N}\to\mathbb{R}$ , and $h\,:\,\mathbb{N}\to\mathbb{R}$ , the notation

(2.1) \begin{equation} G(k,\operatorname{O}\!(g_1(k)),\dots,\operatorname{O}\!(g_d(k)))=\operatorname{O}\!(h(k)),\quad k\in\mathbb{N}, \end{equation}

means that for all $c_1,\dots,c_d\in\mathbb{R}_+$ , there exists a positive real number M such that

\begin{equation*}|G(k, c_1 g_1(k), \dots, c_d g_d(k))|\leq M h(k),\quad k\in\mathbb{N}.\end{equation*}

For example, $\operatorname{O}\!(1)=\operatorname{O}\!(k)$ , $k\in\mathbb{N}$ , because for all $c\in\mathbb{R}_+$ , $|c\cdot1|\leq c\cdot k$ , $k\in\mathbb{N}$ , but we point out that $\operatorname{O}\!(k)=\operatorname{O}\!(1)$ , $k\in\mathbb{N}$ , does not hold. The relation $\operatorname{O}\!(1)=\operatorname{O}\!(k)$ , $k\in\mathbb{N}$ , fits into the framework (2.1) with the following choices: $d=1$ , $G\,:\,\mathbb{R}^2\to \mathbb{R}$ , $G(x,y)=y$ , $(x,y)\in\mathbb{R}^2$ , $g_1\,:\,\mathbb{N}\to\mathbb{R}$ , $g_1(k)=1$ , $k\in\mathbb{N}$ , and $h\,:\,\mathbb{N}\to\mathbb{R}$ , $h(k)=k$ , $k\in\mathbb{N}$ . The relation $f(k) = \operatorname{O}\!(g(k))$ , $k\in\mathbb{N}$ , also fits into the framework (2.1) with the following choices: $d=1$ , $G\,:\,\mathbb{R}^2\to \mathbb{R}$ satisfying $G(k,y)=f(k)$ , $(k,y)\in\mathbb{N}\times\mathbb{R}$ , $g_1$ can be arbitrary, and $h\,:\,\mathbb{N}\to\mathbb{R}$ , $h(k)=g(k)$ , $k\in\mathbb{N}$ ; or with the choices $d=1$ , $G\,:\,\mathbb{R}^2\to \mathbb{R}$ , $G(x,y)=y$ , $(x,y)\in\mathbb{R}^2$ , $g_1\,:\,\mathbb{N}\to\mathbb{R}$ , $g_1(k)\,:\!=\,f(k)$ , $k\in\mathbb{N}$ , and $h\,:\,\mathbb{N}\to\mathbb{R}$ , $h(k)=g(k)$ , $k\in\mathbb{N}$ . For functions $f\,:\,\mathbb{N}\to\mathbb{R}$ and $g\,:\,\mathbb{N}\to\mathbb{R}$ , by the notation $f(k)=\operatorname{O}\!(g(k))\to0$ as $k\to\infty$ , we mean that $f(k)=\operatorname{O}\!(g(k))$ , $k\in\mathbb{N}$ , and $\lim_{k\to\infty}g(k)=0$ , which imply that $\lim_{k\to\infty}f(k)=0$ .

The Euclidean norm on $\mathbb{R}^d$ is denoted by $\Vert\cdot\Vert$ , where $d\in\mathbb{N}$ . The $d\times d$ identity matrix is denoted by ${\boldsymbol{I}}_d$ . For a matrix ${\boldsymbol{B}}\in\mathbb{R}^{d\times d}$ and $m\in\mathbb{Z}_+$ , the (i, j)th entry of ${\boldsymbol{B}}^m$ is denoted by $b_{i,j}^{[m]}$ , where $i,j\in\{1,\ldots,d\}$ . For a function $f\,:\,\mathbb{R}\to\mathbb{R}$ , its positive part is denoted by $f^+$ . Every random variable will be defined on a fixed probability space $(\Omega, {\mathcal A}, \mathbb{P})$ . Convergence in probability, convergence in $L_1$ , convergence almost surely (a.s.), equality in distribution, and almost sure equality are denoted by $\stackrel{\mathbb{P}}{\longrightarrow}$ , $\stackrel{{L_1}}{\longrightarrow}$ , $\stackrel{{\mathrm{a.s.}}}{\longrightarrow}$ , $\stackrel{\mathrm{D}}{=}$ , and $\stackrel{{\mathrm{a.s.}}}{=}$ , respectively. For $d\in\mathbb{N}$ , we will use $\mathsf{D}(\mathbb{R}_+,\mathbb{R}^d)$ for the set of càdlàg functions from $\mathbb{R}_+$ to $\mathbb{R}^d$ , and $\mathsf{C}(\mathbb{R}_+,\mathbb{R}^d)$ for the space of continuous functions from $\mathbb{R}_+$ to $\mathbb{R}^d$ . For a function $f\in\mathsf{D}(\mathbb{R}_+,\mathbb{R}^d)$ , the size of the jump of f at time $t\in\mathbb{R}_+$ is denoted by $\Delta f(t)$ ; that is, $\Delta f(0)\,:\!=\,0$ and $\Delta f (t)\,:\!=\,f(t)-f(t-)$ , $t\in\mathbb{R}_{++}$ , where $f(t-\!)$ is the left-sided limit of f at the point $t\in\mathbb{R}_{++}$ . For $d\in\mathbb{N}$ , the convergence in the sense of the Skorokhod $J_1$ topology of functions in $\mathsf{D}(\mathbb{R}_+,\mathbb{R}^d)$ , the weak convergence of the finite-dimensional distributions of $\mathbb{R}^d$ -valued stochastic processes with sample paths in $\mathsf{D}(\mathbb{R}_+, \mathbb{R}^d)$ , and the weak convergence of $\mathbb{R}^d$ -valued stochastic processes with sample paths in $\mathsf{D}(\mathbb{R}_+, \mathbb{R}^d)$ are denoted by $\stackrel{\textrm{S}_{\textrm{d}}}{\longrightarrow}$ , $\stackrel{\textrm{D}_\textrm{f}}{\longrightarrow}$ , and $\stackrel{\mathrm{D}}{\longrightarrow}$ , respectively (for more details and notation, e.g. for the metric that generates the Skorokhod $J_1$ topology and for the definition of locally uniform (l.u.) convergence, denoted by $\stackrel{{\scriptstyle\mathrm{l.u.}}}{\longrightarrow}$ , see Appendix C).

First, we recall the definition and first-order moment formulae of p-type GWI processes, where $p\in\mathbb{N}$ . For each $k \in \mathbb{Z}_+$ and $i \in \{ 1, \dots, p \}$ , the number of individuals of type i in the kth generation is denoted by $X_{k,i}$ . For simplicity, we suppose that the initial values are $X_{0,i} = 0$ , $i \in \{ 1, \dots, p \}$ . By $\xi_{k,j,i,\ell}$ we denote the number of type $\ell$ offsprings produced by the jth individual who is of type i belonging to the $(k-1)$ th generation. The number of type i immigrants in the kth generation is denoted by $\varepsilon_{k,i}$ . Consider the random vectors

\[ {\boldsymbol{X}}_k \,:\!=\, \begin{bmatrix} X_{k,1} \\ \vdots \\ X_{k,p} \end{bmatrix} , \quad \boldsymbol{\xi}_{k,j,i} \,:\!=\, \begin{bmatrix} \xi_{k,j,i,1} \\ \vdots \\ \xi_{k,j,i,p} \end{bmatrix} , \quad {\boldsymbol{\varepsilon}}_k \,:\!=\, \begin{bmatrix} \varepsilon_{k,1} \\ \vdots \\ \varepsilon_{k,p} \end{bmatrix} . \]

Then we have

(2.2) \begin{equation} {\boldsymbol{X}}_k = \sum_{i=1}^p \sum_{j=1}^{X_{k-1,i}} \boldsymbol{\xi}_{k,j,i} + {\boldsymbol{\varepsilon}}_k , \quad k \in \mathbb{N}, \end{equation}

with ${\boldsymbol{X}}_0={\boldsymbol{0}}$ (and using the convention $\sum_{i=1}^0\,:\!=\,{\boldsymbol{0}}$ ). Here $\big\{\boldsymbol{\xi}_{k,j,i}, \, {\boldsymbol{\varepsilon}}_k\,:\,\: k, j \in \mathbb{N}, \, i \in \{ 1, \dots, p \} \big\}$ are supposed to be independent. Moreover, $\big\{\boldsymbol{\xi}_{k,j,i}\,:\,\: k, j \in \mathbb{N}\big\}$ for each $i \in \{1, \dots, p\}$ , and $\{{\boldsymbol{\varepsilon}}_k\,:\,\: k \in \mathbb{N}\}$ are supposed to consist of identically distributed $\mathbb{Z}_+^p$ -valued random vectors. For notational convenience, let $\{\boldsymbol{\xi}_i\,:\,\: i \in \{1, \ldots, p\}\}$ and ${\boldsymbol{\varepsilon}}$ be random vectors such that $\boldsymbol{\xi}_i \stackrel{\mathrm{D}}{=} \boldsymbol{\xi}_{1,1,i}$ for each $i \in \{1, \ldots, p\}$ and ${\boldsymbol{\varepsilon}} \stackrel{\mathrm{D}}{=} {\boldsymbol{\varepsilon}}_1$ .

In all that follows, we suppose

(2.3) \begin{equation}\mathbb{E}(\|\boldsymbol{\xi}_i\|^2) < \infty, \quad i\in\{ 1,\dots, p\}, \quad\text{and}\quad \mathbb{E} (\|{\boldsymbol{\varepsilon}}\|^2) < \infty.\end{equation}

We introduce the notation

(2.4) \begin{equation} \begin{aligned} {\boldsymbol{A}} \,:\!=\, \begin{bmatrix} \mathbb{E}(\boldsymbol{\xi}_1) &\quad \cdots & \quad \mathbb{E}(\boldsymbol{\xi}_p) \end{bmatrix} \in \mathbb{R}^{p \times p}_+ , \quad {\boldsymbol{b}} \,:\!=\, \mathbb{E}({\boldsymbol{\varepsilon}}) \in \mathbb{R}^{p}_+ , \\ {\boldsymbol{V}}^{(i)} \,:\!=\, \text{var}(\boldsymbol{\xi}_i) \in \mathbb{R}^{p \times p} , \quad i\in\{1,\dots,p\},\quad {\boldsymbol{V}}^{(0)} \,:\!=\, \text{var}({\boldsymbol{\varepsilon}}) \in \mathbb{R}^{p \times p}. \end{aligned} \end{equation}

The matrix ${\boldsymbol{A}}$ and the vector ${\boldsymbol{b}}$ are called the offspring mean matrix and the immigration mean vector, respectively. Note that ${\boldsymbol{A}}=(a_{i,j})_{i,j=1}^p$ , with $a_{i,j} \,:\!=\, \mathbb{E}(\xi_{1,1,j,i})$ , $i,j\in\{1,\ldots,p\}$ , and we mention that some authors define the offspring mean matrix as ${\boldsymbol{A}}^{\top}$ .

For each $k \in \mathbb{Z}_+$ , let ${\mathcal F}_k^{\boldsymbol{X}} \,:\!=\, \sigma({\boldsymbol{X}}_0,\dots, {\boldsymbol{X}}_k)$ , where ${\mathcal F}_0^{\boldsymbol{X}}=\{\emptyset,\Omega\}$ due to ${\boldsymbol{X}}_0={\boldsymbol{0}}$ . By (2.2), we get

(2.5) \begin{equation} \mathbb{E}({\boldsymbol{X}}_k \mid {\mathcal F}_{k-1}^{\boldsymbol{X}}) = \sum_{i=1}^p X_{k-1,i} \mathbb{E}(\boldsymbol{\xi}_i) + {\boldsymbol{b}} = {\boldsymbol{A}} {\boldsymbol{X}}_{k-1} + {\boldsymbol{b}},\quad k\in\mathbb{N} . \end{equation}

Consequently,

\begin{equation*} \mathbb{E}({\boldsymbol{X}}_k) = {\boldsymbol{A}} \mathbb{E}({\boldsymbol{X}}_{k-1}) + {\boldsymbol{b}} , \quad k \in \mathbb{N} , \end{equation*}

and, since ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , we have

\begin{equation*} \mathbb{E}({\boldsymbol{X}}_k) = \sum_{j=0}^{k-1} {\boldsymbol{A}}^j {\boldsymbol{b}} , \quad k \in \mathbb{Z}_+. \end{equation*}

By looking at (2.5), we can interpret ${\boldsymbol{A}}$ as a kind of scaling matrix that determines how many individuals we may expect on average to appear as the offsprings of individuals of the previous generation. It suggests that the offspring mean matrix ${\boldsymbol{A}}$ plays a crucial role in the asymptotic behavior of the sequence $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ . A p-type GWI process $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ is referred to as subcritical, critical, or supercritical, respectively, if $\varrho({\boldsymbol{A}}) < 1$ , $\varrho({\boldsymbol{A}}) = 1$ , or $\varrho({\boldsymbol{A}}) > 1$ , where $\varrho({\boldsymbol{A}})$ denotes the spectral radius of the matrix ${\boldsymbol{A}}$ (see, e.g., [Reference Athreya and Ney1, V.3] or [Reference Quine22]), and it is called indecomposable or decomposable if ${\boldsymbol{A}}$ is irreducible or reducible, respectively. Recall that ${\boldsymbol{A}}$ is called reducible if there exists a permutation matrix ${\boldsymbol{P}}\in\mathbb{R}^{p\times p}$ and $q\in\{1,\ldots,p-1\}$ (and hence p should be at least 2) such that

(2.6) \begin{equation}{\boldsymbol{A}}={\boldsymbol{P}}\begin{bmatrix}{\boldsymbol{R}}& \quad {\boldsymbol{0}}\\[3pt] {\boldsymbol{S}} & \quad {\boldsymbol{T}}\end{bmatrix}{\boldsymbol{P}}^{\top},\end{equation}

where ${\boldsymbol{R}}\in\mathbb{R}^{q\times q},{\boldsymbol{S}}\in\mathbb{R}^{(p-q)\times q},{\boldsymbol{T}}\in\mathbb{R}^{(p-q)\times(p-q)}$ , and ${\boldsymbol{0}}\in\mathbb{R}^{q\times(p-q)}$ is a null matrix. The matrix ${\boldsymbol{A}}$ is called irreducible if it is not reducible (see, e.g., [Reference Horn and Johnson12, Definitions 6.2.21 and 6.2.22]). We emphasize that no $1 \times 1$ matrix is reducible. If a p-type GWI process is decomposable then there exists a decomposition of $\{1,\dots,p\}$ into two disjoint subsets $C_1$ and $C_2$ such that for each $i\in C_1$ and $j\in C_2$ , the individuals of type j cannot produce offspring of type i. If ${\boldsymbol{A}}$ takes the form given in (2.6) with ${\boldsymbol{P}}\,:\!=\,{\boldsymbol{I}}_p$ , then appropriate choices for $C_1$ and $C_2$ are the types corresponding to ${\boldsymbol{R}}$ and ${\boldsymbol{T}}$ , respectively.

We recall a representation of non-negative reducible matrices (see, e.g., [Reference Foster and Ney11]). Let $p\geq2$ , $p\in\mathbb{N}$ , and ${\boldsymbol{A}}\in\mathbb{R}_+^{p\times p}$ . If ${\boldsymbol{A}}$ is reducible then we can find a permutation matrix ${\boldsymbol{P}}_{\boldsymbol{A}}\in\mathbb{R}^{p\times p}$ and $N\in\mathbb{N}$ such that

(2.7) \begin{equation}{\boldsymbol{A}}={\boldsymbol{P}}_{\boldsymbol{A}}\begin{bmatrix}{\boldsymbol{A}}_{1,1} & \quad {\boldsymbol{0}} & \quad \dots & \quad {\boldsymbol{0}}\\[2pt]{\boldsymbol{A}}_{2,1} & \quad {\boldsymbol{A}}_{2,2} & \quad \dots & \quad {\boldsymbol{0}}\\[2pt]\vdots & \quad \vdots & \quad \ddots & \vdots\\[2pt]{\boldsymbol{A}}_{N,1} & \quad {\boldsymbol{A}}_{N,2} & \quad \dots & \quad {\boldsymbol{A}}_{N,N}\end{bmatrix}{\boldsymbol{P}}_{\boldsymbol{A}}^{\top},\end{equation}

where ${\boldsymbol{A}}_{i,j}\in\mathbb{R}^{p_i\times p_j}$ for some $p_i,p_j\in\{1,\dots,p\}$ , $i,j\in\{1,\dots,N\}$ , $p_1+\dots+p_N=p$ , and ${\boldsymbol{A}}_{i,i}$ , $i\in\{1,\dots,N\}$ are all irreducible. This is true when ${\boldsymbol{A}}$ is irreducible as well, as in that case ${\boldsymbol{P}}_{\boldsymbol{A}}={\boldsymbol{I}}_p$ and $N=1$ . For a more detailed argument, see the arXiv version of this paper [Reference Barczy and Bezdány3].

Let $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ be a p-type GWI process such that its offspring mean matrix ${\boldsymbol{A}}$ has the form given in (2.7) with ${\boldsymbol{P}}_{\boldsymbol{A}}\,:\!=\,{\boldsymbol{I}}_p$ . In fact, we can assume that ${\boldsymbol{P}}_{\boldsymbol{A}}$ equals ${\boldsymbol{I}}_p$ , since otherwise instead of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ we can consider the p-type GWI process $({\boldsymbol{P}}_{\boldsymbol{A}}^{\top}{\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ , which has an offspring mean matrix ${\boldsymbol{A}}' = {\boldsymbol{P}}_{\boldsymbol{A}}^{\top}{\boldsymbol{A}}{\boldsymbol{P}}_{\boldsymbol{A}}$ having the form given in (2.7) with ${\boldsymbol{P}}_{{\boldsymbol{A}}'}\,:\!=\,{\boldsymbol{I}}_p$ . Then we can group the types $\{1,\dots,p\}$ into pairwise disjoint sets $C_1,\dots,C_N$ , where $C_1\,:\!=\,\{1,\dots,p_1\}$ and $C_{i+1}\,:\!=\,\{\sum_{j=1}^{i}p_j+1,\dots,\sum_{j=1}^{i}p_j+p_{i+1}\}$ , $i=1,\dots,N-1$ ; that is, $C_i$ consists of the types corresponding to the matrix ${\boldsymbol{A}}_{i,i}$ , $i=1,\ldots,N$ . For $i,j\in\{1,\dots,p\}$ , we say that type j is accessible from type i (in notation $i\to j$ ) if the (j, i)th entry of ${\boldsymbol{A}}^\ell$ is positive for some $\ell\in\mathbb{N}$ . This means that individuals of type i can have descendants of type j. If $i\to j$ and $j\to i$ we say that the types i and j communicate (in notation $i \leftrightarrow j$ ). The relation $\leftrightarrow$ (i.e. communication between types) is an equivalence relation, and the subsets $C_1,\dots,C_N$ of $\{1,\ldots,p\}$ introduced above can be considered as the partition corresponding to $\leftrightarrow$ .

We say that $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ is strongly critical if $\varrho({\boldsymbol{A}}_{i,i})=1$ for each $i\in\{1,\ldots,N\}$ , where ${\boldsymbol{A}}_{i,i}$ , $i\in\{1,\ldots,N\}$ , are the irreducible matrices in the decomposition (2.7) of ${\boldsymbol{A}}$ (see, e.g., [Reference Foster and Ney11]). In this paper, we examine special strongly critical decomposable 3-type GWI processes, with offspring mean matrix ${\boldsymbol{A}}$ given in (2.7) with ${\boldsymbol{P}}_{\boldsymbol{A}}\,:\!=\,{\boldsymbol{I}}_3$ and $N\,:\!=\,3$ . In this case, we have $p_1=p_2=p_3=1$ ; that is, ${\boldsymbol{A}}_{i,i}\,:\!=\,[a_{i,i}]$ , $i\in\{1,2,3\}$ , yielding that ${\boldsymbol{A}}$ is lower triangular, and, since ${\boldsymbol{A}}\in\mathbb{R}_+^{3\times3}$ , we also have $a_{i,i}=|a_{i,i}|=\rho({\boldsymbol{A}}_{i,i})=1$ , $i\in\{1,2,3\}$ . However, we draw attention to the fact that not all the strongly critical decomposable 3-type GWI processes have this type of offspring mean matrix. Studying the asymptotic behavior of 3-type GWI processes with such more general offspring mean matrices could be a topic of future research.

Finally, we note that generalized integer-valued autoregressive (GINAR) processes of order $p\in\mathbb{N}$ (generalizations of integer-valued autoregressive (INAR) processes of order p, first introduced by Latour [Reference Latour20]) are special cases of p-type GWI processes (see, e.g., [Reference Barczy, Nedényi and Pap7, Section 3]). The offspring mean matrix of GINAR processes has a special form; for example, in the case of $p=3$ , its second and third rows are (1,0,0) and (0,1,0), respectively. Consequently, a GINAR process of order 3 such that $a_{1,3} = \mathbb{E}(\xi_{1,1,3,1})>0$ is irreducible (see, e.g., [Reference Barczy, Nedényi and Pap7, Remark 3]), and thus it does not belong to the class of branching processes investigated in this paper. Furthermore, with the use again of Remark 3 in [Reference Barczy, Nedényi and Pap7], the offspring mean matrix of a critical GINAR process of order 3 such that $a_{1,3}=0$ takes the form

\[ \begin{bmatrix} a_{1,1} & \quad a_{1,2} & \quad 0 \\[2pt] 1 & \quad 0 & \quad 0 \\[2pt] 0 & \quad 1 & \quad 0 \\\end{bmatrix}, \quad \text{where $a_{1,1}+a_{1,2}=1$.} \]

This matrix is reducible and has spectral radius 1, yielding that such a GINAR process is critical and decomposable, but it is not strongly critical, and hence it does not belong to the class of branching processes that we investigate.

3. Asymptotic behavior of random step processes formed from some strongly critical decomposable 3-type GWI processes

In what follows, we consider a 3-type Galton–Watson process $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ with immigration starting from ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , and we suppose that the moment conditions (2.3) hold and that the offspring mean matrix ${\boldsymbol{A}}$ is lower triangular with diagonal entries all 1. We point out that we could assume equivalently that ${\boldsymbol{A}}$ is upper triangular with diagonal entries all 1. Indeed, if $([X_{k,1}, X_{k,2}, X_{k,3}]^{\top})_{k\in\mathbb{Z}_+}$ is a 3-type GWI process having an offspring mean matrix with (1, 2)-, (1, 3)-, and (2, 3)-entries 0, then $([X_{k,3},X_{k,2},X_{k,1}]^{\top})_{k\in\mathbb{Z}_+}$ is a 3-type GWI process having an offspring mean matrix with (2, 1)-, (3,1)-, and (3,2)-entries 0. Because of this, Theorems 3.1–3.4 could be reformulated for GWI processes with upper triangular offspring mean matrices (instead of lower triangular ones) as well.

Then the offspring mean matrix ${\boldsymbol{A}}$ and the immigration mean vector ${\boldsymbol{b}}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ take the following forms:

\[ {\boldsymbol{A}} = \begin{bmatrix} \mathbb{E}(\boldsymbol{\xi}_1) &\quad \mathbb{E}(\boldsymbol{\xi}_2) &\quad \mathbb{E}(\boldsymbol{\xi}_3) \end{bmatrix} = \begin{bmatrix} 1 &\quad 0 &\quad 0 \\ a_{2,1} &\quad 1 &\quad 0 \\ a_{3,1} &\quad a_{3,2} &\quad 1 \end{bmatrix} \quad \text{and} \quad {\boldsymbol{b}} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}. \]

Taking into account that $a_{1,2}=a_{1,3}=a_{2,3}=0$ implies that $\xi_{1,1,2,1}\stackrel{{\mathrm{a.s.}}}{=}0$ , $\xi_{1,1,3,1}\stackrel{{\mathrm{a.s.}}}{=}0$ , and $\xi_{1,1,3,2}\stackrel{{\mathrm{a.s.}}}{=}0$ , and (2.2) with $p=3$ takes the form

(3.1) \begin{align} \begin{bmatrix} X_{k,1} \\ X_{k,2} \\ X_{k,3} \end{bmatrix} = \sum_{j=1}^{X_{k-1,1}} \begin{bmatrix} \xi_{k,j,1,1} \\ \xi_{k,j,1,2} \\ \xi_{k,j,1,3} \end{bmatrix} + \sum_{j=1}^{X_{k-1,2}} \begin{bmatrix} 0 \\ \xi_{k,j,2,2} \\ \xi_{k,j,2,3} \end{bmatrix} + \sum_{j=1}^{X_{k-1,3}} \begin{bmatrix} 0 \\ 0 \\ \xi_{k,j,3,3} \end{bmatrix} + \begin{bmatrix} \varepsilon_{k,1} \\ \varepsilon_{k,2} \\ \varepsilon_{k,3} \end{bmatrix} , \quad k \in \mathbb{N} , \end{align}

with $[X_{0,1}, X_{0,2}, X_{0,3}]^{\top}={\boldsymbol{0}}$ . With the notion of accessibility of types of multitype GWI processes (introduced in Section 2), the form of the offspring mean matrix ${\boldsymbol{A}}$ shows that type 1 is not accessible from either type 2 or type 3, and type 2 is not accessible from type 3. In other words (we can also see this from (3.1)), individuals of type 3 can have offsprings only of type 3, and thus they can never have descendants of type 1 or type 2; similarly, individuals of type 2 can have offsprings only of either type 2 or type 3, yielding that they can never have descendants of type 1. We check that it is enough to consider only the following four cases ( $a_{1,2}=a_{1,3}=a_{2,3}=0$ and $a_{1,1}=a_{2,2}=a_{3,3}=1$ for each case):

(3.2)

For abbreviation, we can write the above four cases in matrix form as follows:

\begin{align*} & \begin{bmatrix} 1 & \quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 1 \end{bmatrix}_1 , \quad \begin{bmatrix} 1 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 \\ ++ &\quad + &\quad 1 \end{bmatrix}_2 , \quad \begin{bmatrix} 1 &\quad 0 &\quad 0 \\ ++ &\quad 1 &\quad 0 \\ ++ &\quad 0 &\quad 1 \end{bmatrix}_3 , \quad \begin{bmatrix} 1 &\quad 0 &\quad 0 \\ ++ &\quad 1 &\quad 0 \\ + &\quad ++ &\quad 1 \end{bmatrix}_4 . \end{align*}

To show that it is sufficient to consider only these four cases, consider a 3-type GWI process $([X_{k,1},X_{k,2},X_{k,3}]^{\top})_{k\in\mathbb{Z}_+}$ whose offspring mean matrix is lower triangular with diagonal entries all 1, and assume that this process does not fit into any of the cases listed in (3.2). It is easy to see that (by considering which entries below the diagonal can be 0, and noticing that all but two possibilities are covered by the cases (1)–(4)) such a process must belong to one of the following two cases, (a) or (b), written in matrix form as

\begin{equation*}\begin{bmatrix} 1 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 \\ 0 &\quad ++ &\quad 1 \end{bmatrix}_a,\quad\begin{bmatrix} 1 &\quad 0 &\quad 0 \\ ++ &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 1 \end{bmatrix}_b.\end{equation*}

If $([X_{k,1},X_{k,2},X_{k,3}]^{\top})_{k\in\mathbb{Z}_+}$ belongs to case (a) then the process $([X_{k,2},X_{k,1},X_{k,3}]^{\top})_{k\in\mathbb{Z}_+}$ has an offspring mean matrix with (1, 2)-, (1, 3)-, (2, 1)-, (2, 3)-, and (3, 2)-entries 0, (1, 1)-, (2, 2)-, and (3, 3)-entries 1, and (3, 1)-entry positive, yielding that it belongs to case (2). Similarly, if the process $([X_{k,1},X_{k,2},X_{k,3}]^{\top})_{k\in\mathbb{Z}_+}$ belongs to case (b) then the process $([X_{k,1},X_{k,3},X_{k,2}]^{\top})_{k\in\mathbb{Z}_+}$ has an offspring mean matrix with (1, 2)-, (1, 3)-, (2, 1)-, (2, 3)-, and (3, 2)-entries 0, (1, 1)-, (2, 2)-, and (3, 3)-entries 1, and (3, 1)-entry positive, yielding that it also belongs to case (2).

Note that (3.1) readily yields that the first coordinate process $(X_{k,1})_{k\in\mathbb{Z}_+}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies

\begin{align*} X_{k,1} = \sum_{j=1}^{X_{k-1,1}} \xi_{k,j,1,1} + \varepsilon_{k,1} , \quad k \in \mathbb{N}; \end{align*}

hence, $(X_{k,1})_{k\in\mathbb{Z}_+}$ is a critical single-type GWI process (due to $\mathbb{E}(\xi_{1,1,1,1})=1$ ). Consequently, by a result of Wei and Winnicki [Reference Wei and Winnicki26] (see also Theorem D.2), we have

(3.3) \begin{align} (n^{-1} X_{{\lfloor nt\rfloor},1})_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} ({\mathcal X}_{t,1})_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{align}

where the limit process $({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ is the pathwise unique strong solution of the SDE

\[ \mathrm{d} {\mathcal X}_{t,1} = b_1 \, \mathrm{d} t + \sqrt{v^{(1)}_{1,1} \, {\mathcal X}_{t,1}^+} \, \mathrm{d} {\mathcal W}_{t,1} , \quad t\in\mathbb{R}_+, \quad {\mathcal X}_{0,1} = 0 , \]

where $({\mathcal W}_{t,1})_{t\in\mathbb{R}_+}$ is a standard Wiener process, $b_1= \mathbb{E}(\varepsilon_{1,1})$ , and $v^{(1)}_{1,1}\,:\!=\,\text{var}(\xi_{1,1,1,1})$ . The process $({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ is called a squared Bessel process.

If $a_{2,1}=0$ holds, then $\xi_{1,1,1,2}\stackrel{{\mathrm{a.s.}}}{=}0$ , and (3.1) yields that in this case the second coordinate process $(X_{k,2})_{k\in\mathbb{Z}_+}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies

\begin{align*} X_{k,2} = \sum_{j=1}^{X_{k-1,2}} \xi_{k,j,2,2} + \varepsilon_{k,2} , \quad k \in \mathbb{N} . \end{align*}

Hence, if $a_{2,1}=0$ , then $(X_{k,2})_{k\in\mathbb{Z}_+}$ is a critical single-type GWI process (due to $\mathbb{E}(\xi_{1,1,2,2})=1$ ).

If $a_{3,1}=a_{3,2}=0$ holds, then $\xi_{1,1,1,3}\stackrel{{\mathrm{a.s.}}}{=}0$ and $\xi_{1,1,2,3}\stackrel{{\mathrm{a.s.}}}{=}0$ , and (3.1) yields that in this case the third coordinate process $(X_{k,3})_{k\in\mathbb{Z}_+}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies

\[ X_{k,3} = \sum_{j=1}^{X_{k-1,3}} \xi_{k,j,3,3} + \varepsilon_{k,3} , \quad k \in \mathbb{N} . \]

Hence, if $a_{3,1}=a_{3,2}=0$ holds, then $(X_{k,3})_{k\in\mathbb{Z}_+}$ is a critical single-type GWI process (due to $\mathbb{E}(\xi_{1,1,3,3})=1$ ).

Next we present our results on the asymptotic behavior of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ in the four cases (1)–(4) of its offspring mean matrix ${\boldsymbol{A}}$ . The matrices ${\boldsymbol{V}}^{(i)}$ , $i\in\{0,1,2,3\}$ (introduced in (2.4)), will be written in the form ${\boldsymbol{V}}^{(i)}=(v_{k,\ell}^{(i)})_{k,\ell\in\{1,2,3\}}$ , $i\in\{0,1,2,3\}$ .

Theorem 3.1. Let $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ be a 3-type GWI process such that ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , the moment conditions $\mathbb{E}(\|\boldsymbol{\xi}_i\|^4) < \infty$ , $i\in\{1,2,3\}$ , and $\mathbb{E} (\|{\boldsymbol{\varepsilon}}\|^4) < \infty$ hold, and its offspring mean matrix satisfies (1) of (3.2) (i.e. ${\boldsymbol{A}}={\boldsymbol{I}}_3$ ). Then we have

(3.4) \begin{equation} \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-1} X_{{\lfloor nt\rfloor},2}\\ n^{-1} X_{{\lfloor nt\rfloor},3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2}\\ {\mathcal X}_{t,3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{equation}

where the limit process is the pathwise unique strong solution of the SDE

(3.5) \begin{equation} \begin{cases} \mathrm{d}{\mathcal X}_{t,1} = b_1 \, \mathrm{d} t + \sqrt{v^{(1)}_{1,1} \, {\mathcal X}_{t,1}^+} \, \mathrm{d} {\mathcal W}_{t,1} , \\[1mm] \mathrm{d}{\mathcal X}_{t,2} = b_2 \, \mathrm{d} t + \sqrt{v^{(2)}_{2,2} \, {\mathcal X}_{t,2}^+} \, \mathrm{d} {\mathcal W}_{t,2},\\[1mm] \mathrm{d}{\mathcal X}_{t,3} = b_3 \, \mathrm{d} t + \sqrt{v^{(3)}_{3,3} \, {\mathcal X}_{t,3}^+} \, \mathrm{d} {\mathcal W}_{t,3} , \end{cases} \quad t \in\mathbb{R}_+ , \end{equation}

with initial value $[{\mathcal X}_{0,1} , {\mathcal X}_{0,2} , {\mathcal X}_{0,3}]^{\top} = {\boldsymbol{0}}$ , where $({\mathcal W}_{t,1})_{t\in\mathbb{R}_+}$ , $({\mathcal W}_{t,2})_{t\in\mathbb{R}_+}$ , and $({\mathcal W}_{t,3})_{t\in\mathbb{R}_+}$ are independent standard Wiener processes yielding the independence of $({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ , $({\mathcal X}_{t,2})_{t\in\mathbb{R}_+}$ , and $({\mathcal X}_{t,3})_{t\in\mathbb{R}_+}$ as well.

Theorem 3.2. Let $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ be a 3-type GWI process such that ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , the moment conditions $\mathbb{E}(\|\boldsymbol{\xi}_i\|^4) < \infty$ , $i\in\{1,2\}$ , $\mathbb{E}(\|\boldsymbol{\xi}_3\|^2) < \infty$ , and $\mathbb{E} (\|{\boldsymbol{\varepsilon}}\|^4) < \infty$ hold, and its offspring mean matrix ${\boldsymbol{A}}$ satisfies (2) of (3.2). Then we have

\begin{equation*} \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-1} X_{{\lfloor nt\rfloor},2} \\ n^{-2} X_{{\lfloor nt\rfloor},3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2} \\ {\mathcal X}_{t,3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{equation*}

where the limit process is the pathwise unique strong solution of the SDE

(3.6) \begin{equation} \begin{cases} \mathrm{d}{\mathcal X}_{t,1} = b_1 \, \mathrm{d} t + \sqrt{v^{(1)}_{1,1} {\mathcal X}_{t,1}^+} \, \mathrm{d} {\mathcal W}_{t,1} , \\[2mm] \mathrm{d}{\mathcal X}_{t,2} = b_2 \, \mathrm{d} t + \sqrt{v^{(2)}_{2,2} {\mathcal X}_{t,2}^+} \, \mathrm{d} {\mathcal W}_{t,2} , \\[2mm] \mathrm{d}{\mathcal X}_{t,3} = (a_{3,1} {\mathcal X}_{t,1} + a_{3,2} {\mathcal X}_{t,2}) \, \mathrm{d} t , \end{cases} \quad t \in\mathbb{R}_+, \end{equation}

with initial value $[{\mathcal X}_{0,1}, {\mathcal X}_{0,2},{\mathcal X}_{0,3}]^{\top} = {\boldsymbol{0}}$ , where $({\mathcal W}_{t,1})_{t\in\mathbb{R}_+}$ and $({\mathcal W}_{t,2})_{t\in\mathbb{R}_+}$ are independent standard Wiener processes.

Theorem 3.3. Let $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ be a 3-type GWI process such that ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , the moment condition (2.3) holds, and its offspring mean matrix ${\boldsymbol{A}}$ satisfies (3) of (3.2). Then we have

\begin{equation*} \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-2} X_{{\lfloor nt\rfloor},2} \\ n^{-2} X_{{\lfloor nt\rfloor},3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2} \\ {\mathcal X}_{t,3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{equation*}

where the limit process is the pathwise unique strong solution of the SDE

(3.7) \begin{equation} \begin{cases} \mathrm{d}{\mathcal X}_{t,1} = b_1 \, \mathrm{d} t + \sqrt{v^{(1)}_{1,1} {\mathcal X}_{t,1}^+} \, \mathrm{d} {\mathcal W}_{t,1} , \\[2mm] \mathrm{d}{\mathcal X}_{t,2} = a_{2,1} {\mathcal X}_{t,1} \, \mathrm{d} t , \\[2mm] \mathrm{d}{\mathcal X}_{t,3} = a_{3,1} {\mathcal X}_{t,1} \, \mathrm{d} t , \end{cases} \quad t \in\mathbb{R}_+ , \end{equation}

with initial value $ [{\mathcal X}_{0,1}, {\mathcal X}_{0,2},{\mathcal X}_{0,3}]^{\top} = {\boldsymbol{0}}$ , where $({\mathcal W}_{t,1})_{t\in\mathbb{R}_+}$ is a standard Wiener process.

Theorem 3.4. Let $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ be a 3-type GWI process such that ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , the moment condition (2.3) holds, and its offspring mean matrix ${\boldsymbol{A}}$ satisfies (4) of (3.2). Then we have

\begin{equation*} \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-2} X_{{\lfloor nt\rfloor},2} \\ n^{-3} X_{{\lfloor nt\rfloor},3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2} \\ {\mathcal X}_{t,3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{equation*}

where the limit process is the pathwise unique strong solution of the SDE

(3.8) \begin{equation} \begin{cases} \mathrm{d}{\mathcal X}_{t,1} = b_1 \, \mathrm{d} t + \sqrt{v^{(1)}_{1,1} {\mathcal X}_{t,1}^+} \, \mathrm{d} {\mathcal W}_{t,1} , \\[2mm] \mathrm{d}{\mathcal X}_{t,2} = a_{2,1} {\mathcal X}_{t,1} \, \mathrm{d} t , \\[2mm] \mathrm{d}{\mathcal X}_{t,3} = a_{3,2} {\mathcal X}_{t,2} \, \mathrm{d} t , \end{cases} \quad t \in\mathbb{R}_+, \end{equation}

with initial value $[{\mathcal X}_{0,1}, {\mathcal X}_{0,2},{\mathcal X}_{0,3}]^{\top} = {\boldsymbol{0}}$ , where $({\mathcal W}_{t,1})_{t\in\mathbb{R}_+}$ is a standard Wiener process.

We note that if $b_1=0$ then the limit process in Theorem 3.4 is the identically zero process; that is, $[{\mathcal X}_{t,1},{\mathcal X}_{t,2},{\mathcal X}_{t,3}]^{\top}={\boldsymbol{0}}$ , $t\in\mathbb{R}_+$ . This is in accordance with part (i) of Theorem 4 in [Reference Foster and Ney11] in the special case ${\boldsymbol{a}}_1=b_1=0$ . In the next remark, we rewrite the limit process in Theorem 3.4 in two different forms.

Remark 3.2. The limit process in Theorem 3.4 can be written in the following two forms:

\begin{align*}\begin{bmatrix}{\mathcal X}_{t,1}\\[2pt] {\mathcal X}_{t,2}\\[2pt] {\mathcal X}_{t,3}\end{bmatrix}=\begin{bmatrix}{\mathcal X}_{t,1}\\[2pt] a_{2,1}\int_0^t{\mathcal X}_{s,1}\,\mathrm{d} s\\[2pt] a_{3,2}a_{2,1}\int_0^t\int_0^r{\mathcal X}_{s,1}\,\mathrm{d} s\,\mathrm{d} r\end{bmatrix}&=\begin{bmatrix}{\mathcal X}_{t,1}\\[2pt] a_{2,1}\int_0^t{\mathcal X}_{s,1}\,\mathrm{d} s\\[2pt] a_{3,2}a_{2,1}\int_0^t\int_0^t {{\boldsymbol{1}}}_{[0,r]}(s){\mathcal X}_{s,1}\,\mathrm{d} s\,\mathrm{d} r\end{bmatrix}\\[2pt] &=\begin{bmatrix}{\mathcal X}_{t,1}\\[2pt] a_{2,1}\int_0^t{\mathcal X}_{s,1}\,\mathrm{d} s\\[2pt] a_{3,2}a_{2,1}\int_0^t(t-s){\mathcal X}_{s,1}\,\mathrm{d} s\end{bmatrix}, \quad t\in\mathbb{R}_+,\end{align*}

and

\begin{equation*}\begin{bmatrix}{\mathcal X}_{t,1}\\{\mathcal X}_{t,2}\\{\mathcal X}_{t,3}\end{bmatrix}=\begin{bmatrix}{\mathcal X}_{t,1}\\a_{2,1}\int_0^t(t-s)\,\mathrm{d}{\mathcal X}_{s,1}\\\frac{a_{3,2}a_{2,1}}{2}\int_0^t(t-s)^2\,\mathrm{d}{\mathcal X}_{s,1}\end{bmatrix},\quad t\in\mathbb{R}_+.\end{equation*}

The first formula follows from Fubini’s theorem, and the second one can be checked with the use of Itô’s formula (for details, see the arXiv version of this paper [Reference Barczy and Bezdány3]).

In the next remark, we discuss some connections between Theorems 3.1–3.3 and Theorem 3.4.

In the next remark, we provide a heuristic argument why the 1-fold and 2-fold iterated integral processes of $({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ are expected to appear in the limit process in Theorem 3.4.

Remark 3.4. Let us suppose that the conditions of Theorem 3.4 hold. Then, using (3.1), we can write

\begin{align*} X_{k,2} =& \sum_{j=1}^{X_{k-1,1}}\xi_{k,j,1,2}+\sum_{j=1}^{X_{k-1,2}}\xi_{k,j,2,2}+\varepsilon_{k,2}\\ =&\sum_{j=1}^{X_{k-1,1}}\xi_{k,j,1,2}+\sum_{j=1}^{X_{k-1,2}}\xi_{k,j,2,2}+\varepsilon_{k,2}+X_{k-1,1}(a_{2,1}-a_{2,1})+X_{k-1,2}(1-1)\\[-2pt] =&X_{k-1,2} + a_{2,1}X_{k-1,1} + \sum_{j=1}^{X_{k-1,1}}(\xi_{k,j,1,2}-a_{2,1})+\sum_{j=1}^{X_{k-1,2}}(\xi_{k,j,2,2}-1)+\varepsilon_{k,2} \end{align*}

for each $k\in\mathbb{N}$ . Repeating the same procedure for $X_{k-1,2},\ldots,X_{2,1}$ (or formally, by induction) and using the fact that $X_{0,2}=0$ , we get

\begin{align*} X_{k,2}=&a_{2,1}\sum_{\ell=1}^{k-1}X_{\ell,1}+\sum_{\ell=1}^{k}\sum_{j=1}^{X_{\ell-1,1}}(\xi_{\ell,j,1,2}-a_{2,1}) +\sum_{\ell=1}^{k}\sum_{j=1}^{X_{\ell-1,2}}(\xi_{\ell,j,2,2}-1)+\sum_{\ell=1}^k\varepsilon_{\ell,2}\\ =&a_{2,1}\int_0^{{k}/{n}}nX_{{\lfloor ns\rfloor},1}\,\mathrm{d} s + E_{k,2}, \quad k,n\in\mathbb{N},\end{align*}

where the error term $E_{k,2}$ is defined by

\[ E_{k,2} \,:\!=\,\sum_{\ell=1}^{k}\sum_{j=1}^{X_{\ell-1,1}}(\xi_{\ell,j,1,2}-a_{2,1}) +\sum_{\ell=1}^{k}\sum_{j=1}^{X_{\ell-1,2}}(\xi_{\ell,j,2,2}-1)+\sum_{\ell=1}^k\varepsilon_{\ell,2}, \]

and the last equality follows from (A.1) and the fact that $X_{0,1}=0$ . Similarly, using (3.1), (A.1), and the fact that $X_{0,3}=0$ , we can write

\begin{align*} X_{k,3} = a_{3,2} \int_0^{{k}/{n}} n X_{{\lfloor ns\rfloor},2}\,\mathrm{d} s + E_{k,3}, \quad k,n\in\mathbb{N}, \end{align*}

where the error term $E_{k,3}$ is defined by

\begin{align*} E_{k,3}&\,:\!=\, a_{3,1} \sum_{\ell=1}^{k-1} X_{\ell,1} + \sum_{\ell=1}^k \sum_{j=1}^{X_{\ell-1,1}}(\xi_{\ell,j,1,3}-a_{3,1}) + \sum_{\ell=1}^k \sum_{j=1}^{X_{\ell-1,2}}(\xi_{\ell,j,2,3}-a_{3,2}) \\ &\phantom{\,:\!=\,\,} + \sum_{\ell=1}^k \sum_{j=1}^{X_{\ell-1,3}}(\xi_{\ell,j,3,3}-1) + \sum_{\ell=1}^k \varepsilon_{\ell,1}; \end{align*}

for details, see the arXiv version of this paper [Reference Barczy and Bezdány3]. All in all, writing ${\lfloor nt\rfloor}$ instead of k as well, we can obtain

\begin{align*} \begin{bmatrix} n^{-1}X_{{\lfloor nt\rfloor},1}\\ n^{-2}X_{{\lfloor nt\rfloor},2}\\ n^{-3}X_{{\lfloor nt\rfloor},3} \end{bmatrix} &= \begin{bmatrix} n^{-1}X_{{\lfloor nt\rfloor},1}\\[2pt] a_{2,1}\int_0^{{{\lfloor nt\rfloor}}/{n}}n^{-1}X_{{\lfloor ns\rfloor},1}\,\mathrm{d} s\\[2pt] a_{3,2}\int_0^{{{\lfloor nt\rfloor}}/{n}}n^{-2}X_{{\lfloor nr\rfloor},2}\,\mathrm{d} r \end{bmatrix} + \begin{bmatrix} 0\\[2pt] n^{-2} E_{{\lfloor nt\rfloor},2}\\[2pt] n^{-3} E_{{\lfloor nt\rfloor},3} \end{bmatrix}\\[2pt] & =\begin{bmatrix} n^{-1}X_{{\lfloor nt\rfloor},1}\\[2pt] a_{2,1}\int_0^{{{\lfloor nt\rfloor}}/{n}}n^{-1}X_{{\lfloor ns\rfloor},1}\,\mathrm{d} s\\[2pt] a_{3,2}a_{2,1}\int_0^{{{\lfloor nt\rfloor}}/{n}}\int_0^{{{\lfloor nr\rfloor}}/{n}}n^{-1}X_{{\lfloor ns\rfloor},1}\,\mathrm{d} s\,\mathrm{d} r \end{bmatrix}\\[2pt] &\quad +\begin{bmatrix} 0\\[2pt] n^{-2} E_{{\lfloor nt\rfloor},2}\\[2pt] a_{3,2}\int_0^{{{\lfloor nt\rfloor}}/{n}}n^{-2}E_{{\lfloor nr\rfloor},2}\,\mathrm{d} r+ n^{-3}E_{{\lfloor nt\rfloor},3} \end{bmatrix}\\[2pt] & =\!:\; \widetilde{\boldsymbol{X}}_{{\lfloor nt\rfloor}} + {\boldsymbol{E}}_{\lfloor nt\rfloor}, \quad n\in\mathbb{N}, \; t\in\mathbb{R}_+. \end{align*}

By Theorem D.2, Lemmas C.3 and C.8, and the fact that the limit process in Theorem D.2 has continuous sample paths almost surely, we have $(\widetilde{\boldsymbol{X}}_{\lfloor nt\rfloor})_{t\in\mathbb{R}_+}\stackrel{\mathrm{D}}{\longrightarrow}(\boldsymbol{\mathcal{X}}_t)_{t\in\mathbb{R}_+}$ as $n\to\infty$ , where $(\boldsymbol{\mathcal{X}}_t)_{t\in\mathbb{R}_+}$ is the limit process in Theorem 3.4. Therefore, if the error process $({\boldsymbol{E}}_{\lfloor nt\rfloor})_{t\in\mathbb{R}_+}$ disappeared as $n\to\infty$ in an appropriate sense, then Theorem 3.4 would hold.

4. Preliminaries for the proofs

First, we recall some results about the powers of lower triangular matrices with all the diagonal entries equal to 1.

Lemma 4.1. Let ${\boldsymbol{A}}\in\mathbb{R}^{p\times p}$ be a lower triangular matrix such that $a_{i,i}=1$ , $i\in\{1,\ldots,p\}$ . Then for each $k\in\mathbb{N}$ , we have

(4.1) \begin{equation}{\boldsymbol{A}}^k=\sum_{m=0}^{p-1}\binom{k}{m}\left({\boldsymbol{A}}-{\boldsymbol{I}}_p\right)^m.\end{equation}

Proof. Let ${\boldsymbol{C}} \,:\!=\, {\boldsymbol{A}}-{\boldsymbol{I}}_p$ . Then, by the binomial theorem, we get

\begin{equation*} {\boldsymbol{A}}^k=({\boldsymbol{C}}+{\boldsymbol{I}}_p)^k=\sum_{m=0}^k\binom{k}{m}{\boldsymbol{C}}^m, \quad k\in\mathbb{N}. \end{equation*}

Since ${\boldsymbol{C}}$ is lower triangular and $c_{i,i}=0$ for $i\in\{1,\dots,p\}$ , we have ${\boldsymbol{C}}^m={\boldsymbol{0}}$ whenever $m\geq p$ , $m\in\mathbb{N}$ , so we get

\begin{equation*} {\boldsymbol{A}}^k=\sum_{m=0}^{(p-1)\wedge k}\binom{k}{m}{\boldsymbol{C}}^m, \quad k\in\mathbb{N}. \end{equation*}

This shows (4.1) by our taking into account the convention $\binom{k}{m}=0$ for $k,m\in\mathbb{Z}_+$ with $k<m$ .

In the next remark, we collect a few facts about the powers of the matrix ${\boldsymbol{C}}={\boldsymbol{A}}-{\boldsymbol{I}}_p$ defined in the proof of Lemma 4.1.

Remark 4.1. (i). We check that for each $i,j\in\{1,\dots,p\}$ , we have

(4.2) \begin{equation} c_{i,j}^{[m]}=0 \quad \text{ for $m>i-j$, $m\in\mathbb{N}$.}\end{equation}

First, suppose that $i,j\in\{1,\ldots,p\}$ are such that $i\leq j$ . Then (4.2) holds trivially for $m=1$ , and if it holds for $m\in\mathbb{N}$ then, by the rules of matrix multiplication, the non-negativity of the entries of ${\boldsymbol{C}}$ , and the fact that $c_{i,r}=0$ for $r\geq i$ , $i,r\in\{1,\ldots,p\}$ , we get

\begin{equation*}c_{i,j}^{[m+1]} =\sum_{r=1}^pc_{i,r}c_{r,j}^{[m]}=\sum_{r=1}^{i-1} c_{i,r} c_{r,j}^{[m]}\leq\sum_{r=1}^{j-1} c_{i,r}c_{r,j}^{[m]} =\sum_{r=1}^{j-1}c_{i,r}\cdot 0=0,\end{equation*}

where at the penultimate equality we used the induction hypothesis. Since $c_{i,j}^{[m+1]}\geq 0$ , this implies that $c_{i,j}^{[m+1]}=0$ , and hence we proved (4.2) for $i\leq j$ , $i,j\in\{1,\dots,p\}$ and $m\in\mathbb{N}$ .

Next, suppose that $i,j\in\{1,\ldots,p\}$ are such that $i>j$ . Then $i-j\in\{1,\ldots,p-1\}$ . If $i-j=1$ and $m>i-j$ , $m\in\mathbb{N}$ , then $m-1\in\mathbb{N}$ , and thus

\begin{equation*}c_{i,j}^{[m]}=c_{j+1,j}^{[m]}=\sum_{r=1}^pc_{j+1,r}c_{r,j}^{[m-1]} =\sum_{r=1}^j c_{j+1,r} c_{r,j}^{[m-1]} =\sum_{r=1}^j c_{j+1,r}\cdot0=0,\end{equation*}

where the penultimate equality follows from the previous case. Now suppose that the statement holds whenever $i-j\in\{1,\dots,\ell\}$ with some $\ell\in\{1,\ldots,p-2\}$ . If $i-j=\ell+1$ then for $m\ge \ell+2$ , $m\in\mathbb{N}$ , we have

\begin{equation*} c_{i,j}^{[m]}=\sum_{r=1}^pc_{i,r}c_{r,j}^{[m-1]}=\sum_{r=1}^{i-1}c_{i,r}c_{r,j}^{[m-1]}=\sum_{r=1}^{i-1}c_{i,r}\cdot0=0,\end{equation*}

where the penultimate equality can be checked as follows. If $r\in\{1,\ldots,i-1\}$ is such that $r\leq j$ , then, by the previous case, $c_{r,j}^{[m-1]}=0$ . If $r\in\{1,\ldots,i-1\}$ is such that $r>j$ , then $ 1\leq r-j\leq i-j-1=\ell< \ell+1\leq m-1$ , yielding that $r-j\in\{1,\ldots,\ell\}$ and $m-1> r-j$ . Consequently, by our hypothesis, we have $c_{r,j}^{[m-1]}=0$ .

(ii). We derive that

(4.3) \begin{equation}c_{i,j}^{[i-j]}=\prod_{r=j}^{i -1}a_{r+1,r} \quad \text{for $i>j$, $i,j\in\{1,\dots,p\}$.}\end{equation}

This can be easily shown by induction on the value of $i-j\in\{1,\ldots,p-1\}$ . If $i-j=1$ then $c_{i,j}^{[i-j]}=c_{j+1,j}=a_{j+1,j}$ , so (4.3) holds. Suppose that (4.3) holds whenever $i-j\in\{1,\ldots,\ell\}$ with some $\ell\in\{1,\ldots,p-2\}$ . If $i-j=\ell+1$ then

\begin{equation*} c_{i,j}^{[i-j]}=\sum_{r=1}^pc_{i,r} c_{r,j}^{[\ell]} =\sum_{r=1}^{i-1}a_{i,r} c_{r,j}^{[\ell]}=a_{i,i-1}c_{i-1,j}^{[i-j-1]} =a_{i,i-1}\prod_{r=j}^{i-2}a_{r+1,r} =\prod_{r=j}^{i-1}a_{r+1,r},\end{equation*}

since if $r\in\{1,\ldots,i-2\}$ then $r-j< (i-1)-j=\ell$ and, thus, by (4.2), $c_{r,j}^{[\ell]}=0$ whenever $r\in\{1,\ldots,i-2\}$ .

For a lower triangular matrix ${\boldsymbol{A}}\in\mathbb{R}_+^{p\times p}$ such that $a_{j,j}=1$ , $j\in\{1,\dots,p\}$ , let us recall the notation ${\boldsymbol{C}}={\boldsymbol{A}}-{\boldsymbol{I}}_p$ , and for $i\in\{1,\dots,p\}$ introduce

(4.4) \begin{equation}\eta_i \,:\!=\, \eta_i({\boldsymbol{A}}) \,:\!=\,\max_{m\in\{1,\dots,p\}}\{m \mid \exists\, j\in\{1,\dots,p\}\;:\; c_{i,j}^{[m-1]}>0\}, \end{equation}

where we recall that $c_{i,j}^{[m-1]}$ denotes the (i, j)th entry of ${\boldsymbol{C}}^{m-1} =({\boldsymbol{A}}-{\boldsymbol{I}}_p)^{m-1}$ . Note that because $c_{i,i}^{[0]}=({\boldsymbol{I}}_p)_{i,i}=1$ , we have $\eta_i\geq1$ for each $i\in\{1,\dots,p\}$ .

In the next lemma, we derive an inequality between $\eta_i$ and $\eta_j$ , $i,j\in\{1,\dots,p\}$ .

Proof. Assume $c_{i,j}^{[m]}>0$ for some $i,j\in\{1,\ldots,p\}$ and $m\in\{0,\dots,p-1\}$ . Since $\eta_j\geq 1$ , by definition (4.4), there exists some $r_0\in\{1,\dots,p\}$ such that $c_{j,r_0}^{[\eta_j-1]}>0$ , and, by the properties of matrix multiplication and the non-negativity of the coefficients $c_{\ell,k}^{[q]}$ , $q\in\mathbb{Z}_+$ , $\ell,k\in\{1,\ldots,p\}$ , we get

\begin{equation*} c_{i,r_0}^{[\eta_j+m-1]}=\sum_{r=1}^pc_{i,r}^{[m]}c_{r,r_0}^{[\eta_j-1]}\geq c_{i,j}^{[m]}c_{j,r_0}^{[\eta_j-1]}>0. \end{equation*}

This implies $\eta_i\geq\eta_j+m$ .

In the rest of this section, we consider only the case $p=3$ . First, we specialize (4.1) to the case $p=3$ .

Corollary 4.1. Let ${\boldsymbol{A}}\in\mathbb{R}^{3\times 3}$ be a lower triangular matrix such that $a_{i,i}=1$ , $i\in\{1,2,3\}$ . Then for each $k\in\mathbb{Z}_+$ , we have

\[ {\boldsymbol{A}}^k =\begin{bmatrix} 1 &\quad 0 &\quad 0 \\ a_{2,1} &\quad 1 &\quad 0 \\ a_{3,1} &\quad a_{3,2} &\quad 1 \\ \end{bmatrix}^k = \begin{bmatrix} 1 &\quad 0 &\quad 0 \\ k a_{2,1} &\quad 1 &\quad 0 \\ \binom{k}{2}a_{3,2}a_{2,1} + ka_{3,1}&\quad ka_{3,2} &\quad 1 \\ \end{bmatrix}. \]

.

Proof. In the considered case, with ${\boldsymbol{C}}={\boldsymbol{A}}-{\boldsymbol{I}}_3$ , we have

(4.5) \begin{equation}{\boldsymbol{C}}^0=\begin{bmatrix} 1 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 1 \\ \end{bmatrix}, \quad{\boldsymbol{C}}^1= \begin{bmatrix} 0 &\quad 0 &\quad 0 \\ a_{2,1} &\quad 0 &\quad 0 \\ a_{3,1} &\quad a_{3,2} &\quad 0 \\ \end{bmatrix},\quad{\boldsymbol{C}}^2= \begin{bmatrix} 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 \\ a_{3,2}a_{2,1} &\quad 0 &\quad 0 \\ \end{bmatrix},\end{equation}

and, by (4.1), for each $k\in\mathbb{N}$ we have

\begin{equation*}\begin{bmatrix} 1 &\quad 0 &\quad 0 \\ a_{2,1} &\quad 1 &\quad 0 \\ a_{3,1} &\quad a_{3,2} &\quad 1 \\ \end{bmatrix}^k=\binom{k}{0}{\boldsymbol{C}}^0+\binom{k}{1}{\boldsymbol{C}}^1+\binom{k}{2}{\boldsymbol{C}}^2,\end{equation*}

yielding the assertion.

The quantities $\eta_i$ , $i\in\{1,2,3\}$ , defined in (4.4), will become relevant in the moment estimations presented in Lemmas B.2 and B.4, which play a crucial role in the proofs of Theorems 3.1–3.4, respectively. Next, we calculate their exact values in the cases (1)–(4) of (3.2).

Lemma 4.3. Let $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ be a strongly critical 3-type GWI process such that ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , the moment condition (2.3) holds, and suppose the offspring mean matrix ${\boldsymbol{A}}$ is lower triangular such that $a_{i,i}=1$ , $i\in\{1,2,3\}$ . Then in the four cases (1)–(4) of (3.2), the quantities $\eta_i$ , $i\in\{1,2,3\}$ , defined in (4.4), take the following values:

The explicit form of ${\boldsymbol{A}}^k$ , $k\in\mathbb{N}$ , in Corollary 4.1 together with (B.1) enables us to describe the asymptotic behavior of $\mathbb{E}({\boldsymbol{X}}_k)$ as $k\to\infty$ ; see the next proposition.

Proposition 4.1. Let $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ be a 3-type GWI process such that ${\boldsymbol{X}}_0={\boldsymbol{0}}$ , and the moment condition (2.3) holds. If the offspring mean matrix ${\boldsymbol{A}}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies (1) of (3.2), then we have

(4.6) \begin{align} \lim_{k\to\infty} \begin{bmatrix} k^{-1} \mathbb{E}(X_{k,1}) \\ k^{-1} \mathbb{E}(X_{k,2}) \\ k^{-1} \mathbb{E}(X_{k,3}) \\ \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ \end{bmatrix}. \end{align}

If the offspring mean matrix ${\boldsymbol{A}}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies (2) of (3.2), then we have

(4.7) \begin{align} \lim_{k\to\infty} \begin{bmatrix} k^{-1} \mathbb{E}(X_{k,1}) \\ k^{-1} \mathbb{E}(X_{k,2}) \\ k^{-2} \mathbb{E}(X_{k,3}) \\ \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ \frac{1}{2}(b_1a_{3,1} + b_2a_{3,2}) \\ \end{bmatrix}. \end{align}

If the offspring mean matrix ${\boldsymbol{A}}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies (3) of (3.2), then we have

(4.8) \begin{align} \lim_{k\to\infty} \begin{bmatrix} k^{-1} \mathbb{E}(X_{k,1}) \\ k^{-2} \mathbb{E}(X_{k,2}) \\ k^{-2} \mathbb{E}(X_{k,3}) \\ \end{bmatrix} = \begin{bmatrix} b_1 \\ \frac{1}{2}b_1a_{2,1} \\[2pt] \frac{1}{2}b_1a_{3,1} \\ \end{bmatrix}. \end{align}

If the offspring mean matrix ${\boldsymbol{A}}$ of $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies (4) of (3.2), then we have

(4.9) \begin{align} \lim_{k\to\infty} \begin{bmatrix} k^{-1} \mathbb{E}(X_{k,1}) \\ k^{-2} \mathbb{E}(X_{k,2}) \\ k^{-3} \mathbb{E}(X_{k,3}) \\ \end{bmatrix} = \begin{bmatrix} b_1 \\ \frac{1}{2}b_1a_{2,1} \\[2pt] \frac{1}{6}b_1a_{3,2}a_{2,1} \\ \end{bmatrix}. \end{align}

Proof. By (B.1), we have $\mathbb{E}({\boldsymbol{X}}_k)= \sum_{j=0}^{k-1} {\boldsymbol{A}}^j {\boldsymbol{b}}$ , $k\in\mathbb{N}$ . With the use of Corollary 4.1, this implies that

\begin{align*} \mathbb{E}({\boldsymbol{X}}_k) & = \begin{bmatrix} k &\quad 0 &\quad 0 \\ a_{2,1}\sum_{j=1}^{k-1} j &\quad k &\quad 0 \\ a_{3,2}a_{2,1} \sum_{j=1}^{k-1} \binom{j}{2} + a_{3,1}\sum_{j=1}^{k-1} j &\quad a_{3,2}\sum_{j=1}^{k-1} j &\quad k \\ \end{bmatrix} {\boldsymbol{b}} \\ & = \begin{bmatrix} k &\quad 0 &\quad 0 \\ a_{2,1}\frac{(k-1)k}{2} &\quad k &\quad 0 \\ \frac{a_{3,2}a_{2,1}}{2} \sum_{j=1}^{k-1} (j^2-j) + a_{3,1}\frac{(k-1)k}{2} &\quad a_{3,2}\frac{(k-1)k}{2} &\quad k \\ \end{bmatrix} {\boldsymbol{b}}, \quad k\in\mathbb{N}, \end{align*}

where

\[ \sum_{j=1}^{k-1} (j^2-j) = \frac{(k-1)k(2(k-1)+1)}{6} - \frac{(k-1)k}{2} = \frac{(k-2)(k-1)k}{3}. \]

This yields that

(4.10) \begin{align} \mathbb{E}({\boldsymbol{X}}_k) = \begin{bmatrix} b_1k \\ b_1a_{2,1}\binom{k}{2} + b_2k \\ b_1a_{3,2}a_{2,1}\binom{k}{3} + (b_1a_{3,1} + b_2a_{3,2})\binom{k}{2} + b_3k \\ \end{bmatrix}, \quad k\in\mathbb{N}. \end{align}

Hence (4.6)–(4.9) follow by simple computation (for details, see the arXiv version of this paper [Reference Barczy and Bezdány3]).

We note that Proposition 4.1 is a generalization of Theorem 3 in [Reference Foster and Ney11] in the case of $p=3$ ; for more details, see the last paragraph of Remark B.3 of the arXiv version of this paper [Reference Barczy and Bezdány3]. We remark also that the normalization of the coordinates of $\mathbb{E}({\boldsymbol{X}}_k)$ in Proposition 4.1 in the cases (1)–(4) of (3.2) are exactly the same ones as the normalizations for the coordinates of ${\boldsymbol{X}}_{{\lfloor nt\rfloor}}$ in Theorems 3.1–3.4 corresponding to the cases (1)–(4) of (3.2). For the order of the moments of a strongly critical p-type GWI process starting from ${\boldsymbol{0}}$ and with lower triangular offspring mean matrix having diagonal entries 1, see Lemma B.2 as well.

Next we derive a decomposition for the 3-type GWI process $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ starting from ${\boldsymbol{0}}$ , satisfying the moment conditions (2.3), and having a lower triangular offspring mean matrix ${\boldsymbol{A}}$ with diagonal entries all 1. Let us introduce the sequence

(4.11) \begin{equation} \begin{split} \begin{bmatrix} M_{k,1} \\ M_{k,2} \\ M_{k,3} \end{bmatrix} &\,:\!=\, {\boldsymbol{M}}_k \,:\!=\, {\boldsymbol{X}}_k - \mathbb{E}\big({\boldsymbol{X}}_k \mid {\mathcal F}_{k-1}^{{\boldsymbol{X}}}\big) = {\boldsymbol{X}}_k - {\boldsymbol{A}} {\boldsymbol{X}}_{k-1} - {\boldsymbol{b}} \\ &= \begin{bmatrix} X_{k,1} - X_{k-1,1} - b_1 \\ X_{k,2} - a_{2,1} X_{k-1,1} - X_{k-1,2} - b_2 \\ X_{k,3} - a_{3,1} X_{k-1,1} - a_{3,2} X_{k-1,2} - X_{k-1,3} - b_3 \end{bmatrix} , \quad k \in \mathbb{N} , \end{split} \end{equation}

of martingale differences with respect to the filtration $({\mathcal F}_k^{\boldsymbol{X}})_{k\in\mathbb{Z}_+}$ , where we used (2.5). From (4.11), we obtain the recursion $ {\boldsymbol{X}}_k = {\boldsymbol{A}} {\boldsymbol{X}}_{k-1} + {\boldsymbol{M}}_k + {\boldsymbol{b}}$ , $k \in \mathbb{N}$ , which together with ${\boldsymbol{X}}_0={\boldsymbol{0}}$ implies

(4.12) \begin{equation} {\boldsymbol{X}}_k = \sum_{\ell=1}^k {\boldsymbol{A}}^{k-\ell} ({\boldsymbol{M}}_\ell + {\boldsymbol{b}}) , \quad k \in \mathbb{N}. \end{equation}

Finally, by (4.12) and Corollary 4.1, we can get a decomposition

(4.13) \begin{equation} \begin{bmatrix} X_{k,1} \\ X_{k,2} \\ X_{k,3} \end{bmatrix} = \begin{bmatrix} X_{k,1}^{(1)} \\ a_{2,1} X_{k,2}^{(1)} + X_{k,2}^{(2)} \\ a_{3,2}a_{2,1} X_{k,3}^{(1)} + a_{3,1} X_{k,3}^{(2)} + a_{3,2} X_{k,3}^{(3)} + X_{k,3}^{(4)} \end{bmatrix} , \quad k \in \mathbb{N} , \end{equation}

where the stochastic processes $(X_{k,1}^{(1)})_{k\in\mathbb{N}}$ , $(X_{k,2}^{(1)})_{k\in\mathbb{N}}$ , $(X_{k,2}^{(2)})_{k\in\mathbb{N}}$ , $(X_{k,3}^{(1)})_{k\in\mathbb{N}}$ , $(X_{k,3}^{(2)})_{k\in\mathbb{N}}$ , $(X_{k,3}^{(3)})_{k\in\mathbb{N}}$ , and $(X_{k,3}^{(4)})_{k\in\mathbb{N}}$ are given by

\begin{align*} X_{k,1}^{(1)} \,:\!=\, \sum_{\ell=1}^k (M_{\ell,1} + b_1), \quad X_{k,2}^{(1)} \,:\!=\, \sum_{\ell=1}^k (k-\ell) (M_{\ell,1} + b_1), \quad X_{k,2}^{(2)} \,:\!=\, \sum_{\ell=1}^k (M_{\ell,2} + b_2), \end{align*}

\begin{align*} & X_{k,3}^{(1)} \,:\!=\, \frac{1}{2} \sum_{\ell=1}^k (k-\ell)(k-\ell-1) (M_{\ell,1} + b_1) ,\quad X_{k,3}^{(2)} \,:\!=\, \sum_{\ell=1}^k (k-\ell) (M_{\ell,1} + b_1) = X_{k,2}^{(1)},\\[1mm] & X_{k,3}^{(3)} \,:\!=\, \sum_{\ell=1}^k (k-\ell)(M_{\ell,2} + b_2),\quad \mbox{and} \quad X_{k,3}^{(4)} \,:\!=\, \sum_{\ell=1}^k (M_{\ell,3} + b_3). \end{align*}

Indeed, for each $k\in\mathbb{N}$ , we have

\begin{align*} &X_{k,1} = \sum_{\ell=1}^k ({\boldsymbol{A}}^{k-\ell})_{1,1} (M_{\ell,1} + b_1) = \sum_{\ell=1}^k (M_{\ell,1} + b_1), \\ &X_{k,2} = \sum_{\ell=1}^k ({\boldsymbol{A}}^{k-\ell})_{2,1} (M_{\ell,1} + b_1) + \sum_{\ell=1}^k ({\boldsymbol{A}}^{k-\ell})_{2,2} (M_{\ell,2} + b_2)\\ &\phantom{X_{k,2}} = a_{2,1} \sum_{\ell=1}^k (k-\ell) (M_{\ell,1} + b_1) + \sum_{\ell=1}^k (M_{\ell,2} + b_2) \\ &\phantom{X_{k,2}} = a_{2,1} X_{k,2}^{(1)} + X_{k,2}^{(2)} \end{align*}

and

\begin{align*} &X_{k,3} = \sum_{\ell=1}^k ({\boldsymbol{A}}^{k-\ell})_{3,1} (M_{\ell,1} + b_1) + \sum_{\ell=1}^k ({\boldsymbol{A}}^{k-\ell})_{3,2} (M_{\ell,2} + b_2) + \sum_{\ell=1}^k ({\boldsymbol{A}}^{k-\ell})_{3,3} (M_{\ell,3} + b_3)\\ &\phantom{X_{k,3}} = \sum_{\ell=1}^k \left( \binom{k-\ell}{2}a_{3,2}a_{2,1} + (k-\ell)a_{3,1} \right) (M_{\ell,1} + b_1) \\ &\phantom{X_{k,3}=\;} + \sum_{\ell=1}^k (k-\ell) a_{3,2} (M_{\ell,2} + b_2) + \sum_{\ell=1}^k (M_{\ell,3} + b_3)\\ &\phantom{X_{k,3}} = a_{3,2}a_{2,1} \sum_{\ell=1}^k \binom{k-\ell}{2} (M_{\ell,1} + b_1) + a_{3,1} \sum_{\ell=1}^k (k-\ell)(M_{\ell,1} + b_1) \\ &\phantom{X_{k,3}=\;} + a_{3,2} \sum_{\ell=1}^k (k-\ell)(M_{\ell,2} + b_2) + \sum_{\ell=1}^k (M_{\ell,3} + b_3), \end{align*}

and thus we have

\begin{align*}X_{k,3} = a_{3,2}a_{2,1} X_{k,3}^{(1)} + a_{3,1} X_{k,2}^{(1)} + a_{3,2}X_{k,3}^{(3)} + X_{k,3}^{(4)}, \quad k\in\mathbb{N}.\end{align*}

Using that $X_{j,1}^{(1)}=\sum_{\ell=1}^j (M_{\ell,1} + b_1)$ and $X_{j,2}^{(2)}=\sum_{\ell=1}^j (M_{\ell,2} + b_2)$ , $j\in\mathbb{N}$ , by Lemma A.1, for each $k,n\in\mathbb{N}$ , we have

(4.14) \begin{align}\begin{split}X_{k,2}^{(1)}&=X_{k,3}^{(2)} = \sum_{\ell=1}^k (k - \ell) (M_{\ell,1} + b_1) =n\int_0^{{k}/{n}}X_{{\lfloor ns\rfloor},1}^{(1)}\,\mathrm{d} s,\\X_{k,3}^{(3)} &= \sum_{\ell=1}^k (k - \ell) (M_{\ell,2} + b_2) =n\int_0^{{k}/{n}}X_{{\lfloor ns\rfloor},2}^{(2)}\,\mathrm{d} s,\\X_{k,3}^{(1)} &=\sum_{\ell=1}^k \binom{k-\ell}{2} (M_{\ell,1} + b_1) =n^2\int_0^{{k}/{n}}\left(\int_0^{{{\lfloor nr\rfloor}}/{n}}X_{{\lfloor ns\rfloor},1}^{(1)}\,\mathrm{d} s\right)\,\mathrm{d} r.\end{split}\end{align}

5. Proof of Theorem 3.1

The proof is analogous to the proof of Theorem 2.1 in [Reference Barczy, Bezdány and Pap5], but, for completeness, we give a detailed proof. We divide the proof into several steps.

Step 1(a). The SDE (3.5) has a pathwise unique strong solution $(\boldsymbol{\mathcal{X}}_t\,:\!=\,[{\mathcal X}_{t,1},{\mathcal X}_{t,2},{\mathcal X}_{t,3}]^{\top} )_{t\in\mathbb{R}_+}$ for all initial values $\boldsymbol{\mathcal{X}}_0 = {\boldsymbol{x}} \in \mathbb{R}^3$ , and if ${\boldsymbol{x}} \in \mathbb{R}_+^3$ then $\boldsymbol{\mathcal{X}}_t \in \mathbb{R}_+^3$ almost surely for all $t \in \mathbb{R}_+$ , since $b_i$ , $v^{(i)}_{i,i}\in \mathbb{R}_+$ , $i=1,2,3$ (see, e.g., [Reference Ikeda and Watanabe13, Chapter IV, Example 8.2]). By (3.3), we have $(n^{-1} X_{{\lfloor nt\rfloor},1})_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} ({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ as $n \to \infty$ , where $({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ satisfies the first equation of the SDE (3.5) with initial value ${\mathcal X}_{0,1}=0$ . Similarly, since $a_{2,1}=a_{3,1}=a_{3,2}=0$ , as explained in Section 3, the second and third coordinate processes $(X_{k,2})_{k\in\mathbb{Z}_+}$ and $(X_{k,3})_{k\in\mathbb{Z}_+}$ are critical single-type GWI processes, so $(n^{-1} X_{{\lfloor nt\rfloor},2})_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} ({\mathcal X}_{t,2})_{t\in\mathbb{R}_+}$ as $n \to \infty$ , and $(n^{-1} X_{{\lfloor nt\rfloor},3})_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} ({\mathcal X}_{t,3})_{t\in\mathbb{R}_+}$ as $n \to \infty$ , where $({\mathcal X}_{t,2})_{t\in\mathbb{R}_+}$ and $({\mathcal X}_{t,3})_{t\in\mathbb{R}_+}$ satisfy the second and third equations of the SDE (3.5) with initial values ${\mathcal X}_{0,2}=0$ and ${\mathcal X}_{0,3}=0$ , respectively. However, we need to prove joint convergence of $(n^{-1} X_{{\lfloor nt\rfloor},1})_{t\in\mathbb{R}_+}$ , $(n^{-1} X_{{\lfloor nt\rfloor},2})_{t\in\mathbb{R}_+}$ and $(n^{-1} X_{{\lfloor nt\rfloor},3})_{t\in\mathbb{R}_+}$ as $n\to\infty$ .

Step 1(b). Using $a_{2,1}=a_{3,1}=a_{3,2}=0$ and (4.11), we find that the sequence $({\boldsymbol{M}}_k)_{k\in\mathbb{N}}$ of martingale differences with respect to the filtration $({\mathcal F}_k^{\boldsymbol{X}})_{k\in\mathbb{Z}_+}$ takes the form

\[ {\boldsymbol{M}}_k = {\boldsymbol{X}}_k - {\boldsymbol{X}}_{k-1} - {\boldsymbol{b}} , \quad k \in \mathbb{N} . \]

Consider the random step processes

(5.1) \begin{align} \boldsymbol{\mathcal{M}}_t^{(n)} \,:\!=\, \begin{bmatrix} {\mathcal M}_{t,1}^{(n)} \\ {\mathcal M}_{t,2}^{(n)} \\ {\mathcal M}_{t,3}^{(n)} \end{bmatrix} \,:\!=\, \frac{1}{n} \sum_{k=1}^{\lfloor nt\rfloor} {\boldsymbol{M}}_k = \frac{1}{n} {\boldsymbol{X}}_{\lfloor nt\rfloor} - \frac{{\lfloor nt\rfloor}}{n} {\boldsymbol{b}} , \quad t \in \mathbb{R}_+ , \quad n \in \mathbb{N} , \end{align}

where at the third equality we used that ${\boldsymbol{X}}_0 = {\boldsymbol{0}}$ . We show that

(5.2) \begin{equation} (\boldsymbol{\mathcal{M}}_t^{(n)})_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} (\boldsymbol{\mathcal{M}}_t)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{equation}

where the limit process $\boldsymbol{\mathcal{M}}_t = ([{\mathcal M}_{t,1}, {\mathcal M}_{t,2}, {\mathcal M}_{t,3}]^{\top})_{t\in\mathbb{R}_+}$ is the pathwise unique strong solution of the SDE

(5.3) \begin{equation} \begin{cases} \mathrm{d} {\mathcal M}_{t,1} = \sqrt{v^{(1)}_{1,1} ({\mathcal M}_{t,1} + b_1 t)^+ } \, \mathrm{d} {\mathcal W}_{t,1} , \quad t\in\mathbb{R}_+,\\[1mm] \mathrm{d} {\mathcal M}_{t,2} = \sqrt{v^{(2)}_{2,2} ({\mathcal M}_{t,2} + b_2 t)^+ } \, \mathrm{d} {\mathcal W}_{t,2} , \quad t\in\mathbb{R}_+,\\[1mm] \mathrm{d} {\mathcal M}_{t,3} = \sqrt{v^{(3)}_{3,3} ({\mathcal M}_{t,3} + b_3 t)^+ } \, \mathrm{d} {\mathcal W}_{t,3} , \quad t\in\mathbb{R}_+, \end{cases} \end{equation}

with initial value $ \boldsymbol{\mathcal{M}}_0 = {\boldsymbol{0}}$ , where $({\mathcal W}_{t,1})_{t\in\mathbb{R}_+}$ , $({\mathcal W}_{t,2})_{t\in\mathbb{R}_+}$ , and $({\mathcal W}_{t,3})_{t\in\mathbb{R}_+}$ are independent standard Wiener processes.

Step 1(c). We check that the SDE (5.3) has a pathwise unique strong solution $(\boldsymbol{\mathcal{M}}_t)_{t\in\mathbb{R}_+}$ for all initial values $\boldsymbol{\mathcal{M}}_0 = {\boldsymbol{x}} \in \mathbb{R}^3$ . Observe that if $(\boldsymbol{\mathcal{M}}_t)_{t\in\mathbb{R}_+}$ is a strong solution of the SDE (5.3) with initial value $ \boldsymbol{\mathcal{M}}_0 = {\boldsymbol{x}} \in \mathbb{R}^3$ , then, by Itô’s formula, the process $[{\mathcal P}_{t,1}, {\mathcal P}_{t,2} , {\mathcal P}_{t,3}]^{\top} \,:\!=\, \boldsymbol{\mathcal{M}}_t + {\boldsymbol{b}} t$ , $t \in \mathbb{R}_+$ , is a pathwise unique strong solution of the SDE

(5.4) \begin{equation} \begin{cases} \mathrm{d} {\mathcal P}_{t,1} = b_1 \, \mathrm{d} t + \sqrt{v^{(1)}_{1,1} \, {\mathcal P}_{t,1}^+} \, \mathrm{d} {\mathcal W}_{t,1} , \quad t\in\mathbb{R}_+, \\[1mm] \mathrm{d} {\mathcal P}_{t,2} = b_2 \, \mathrm{d} t + \sqrt{v^{(2)}_{2,2} \, {\mathcal P}_{t,2}^+} \, \mathrm{d} {\mathcal W}_{t,2}, \quad t\in\mathbb{R}_+,\\[1mm] \mathrm{d} {\mathcal P}_{t,3} = b_3 \, \mathrm{d} t + \sqrt{v^{(3)}_{3,3} \, {\mathcal P}_{t,3}^+} \, \mathrm{d} {\mathcal W}_{t,3}, \quad t\in\mathbb{R}_+, \end{cases} \end{equation}

with initial value $[{\mathcal P}_{0,1}, {\mathcal P}_{0,2}, {\mathcal P}_{0,3}]^{\top} = {\boldsymbol{x}}$ . Conversely, if $[{\mathcal P}_{t,1}, {\mathcal P}_{t,2}, {\mathcal P}_{t,3}]^{\top}$ , $t \in \mathbb{R}_+$ , is a strong solution of the SDE (5.4) with initial value $[{\mathcal P}_{0,1}, {\mathcal P}_{0,2}, {\mathcal P}_{0,3}]^{\top} = {\boldsymbol{p}} \in \mathbb{R}^3$ , then, by Itô’s formula, the process $\boldsymbol{\mathcal{M}}_t \,:\!=\, [{\mathcal P}_{t,1}, {\mathcal P}_{t,2}, {\mathcal P}_{t,3}]^{\top} - {\boldsymbol{b}} t$ , $t \in \mathbb{R}_+$ , is a strong solution of the SDE (5.3) with initial value $\boldsymbol{\mathcal{M}}_0 = {\boldsymbol{p}}$ . The equations in (5.4) are the same as in (3.5); thus, as explained in step 1(a), the SDE (5.4) admits a unique strong solution with arbitrary initial value in $\mathbb{R}^3$ . Consequently, the SDE (5.3) (with initial value ${\boldsymbol{0}}\in\mathbb{R}^3$ ) admits a unique strong solution $(\boldsymbol{\mathcal{M}}_t)_{t\in\mathbb{R}_+}$ , and $(\boldsymbol{\mathcal{M}}_t + {\boldsymbol{b}} t)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{=} (\boldsymbol{\mathcal{X}}_t)_{t\in\mathbb{R}_+}$ .

Step 2(a). To prove (5.2), we want to apply Theorem D.1 with the following choices: $d = r = 3$ , $\boldsymbol{\mathcal{U}} = \boldsymbol{\mathcal{M}}$ , ${\boldsymbol{U}}_k^{(n)} = n^{-1} {\boldsymbol{M}}_k$ , $n,k\in\mathbb{N}$ , ${\boldsymbol{U}}_0^{(n)} ={\boldsymbol{0}}$ , $n\in\mathbb{N}$ , ${\mathcal F}_k^{(n)} = {\mathcal F}_k^{\boldsymbol{X}}$ , $n \in \mathbb{N}$ , $k \in \mathbb{Z}_+$ (yielding $\boldsymbol{\mathcal{U}}^{(n)} = \boldsymbol{\mathcal{M}}^{(n)}$ , $n\in\mathbb{N}$ ), and $\boldsymbol{\beta} \,:\, \mathbb{R}_+ \times \mathbb{R}^3 \to \mathbb{R}^3$ and $\boldsymbol{\gamma} \,:\, \mathbb{R}_+ \times \mathbb{R}^3 \to \mathbb{R}^{3\times3}$ given by $\boldsymbol{\beta}(t, {\boldsymbol{x}}) = {\boldsymbol{0}}$ and

\[ \boldsymbol{\gamma}(t, {\boldsymbol{x}}) = \begin{bmatrix} \sqrt{v^{(1)}_{1,1}(x_1 + b_1 t)^+} & 0 & 0 \\ 0 & \sqrt{v^{(2)}_{2,2}(x_2 + b_2 t)^+} & 0 \\ 0 & 0 & \sqrt{v^{(3)}_{3,3}(x_3 + b_3 t)^+} \\ \end{bmatrix} \]

for $t \in \mathbb{R}_+$ and ${\boldsymbol{x}} = [x_1, x_2,x_3]^{\top} \in \mathbb{R}^3$ . With these notations, the SDE (5.3) takes the form

\begin{equation*}\mathrm{d}\boldsymbol{\mathcal{M}}_t=\boldsymbol{\beta}(t,\boldsymbol{\mathcal{M}}_t)+\boldsymbol{\gamma}(t,\boldsymbol{\mathcal{M}}_t)\mathrm{d}\boldsymbol{\mathcal{W}}_t,\quad t\in\mathbb{R}_+,\end{equation*}

with initial value $\boldsymbol{\mathcal{M}}_0={\boldsymbol{0}}\in\mathbb{R}^3$ .

The convergence ${\boldsymbol{U}}_0^{(n)}\stackrel{\mathrm{D}}{\longrightarrow} {\boldsymbol{0}}$ as $n\to\infty$ , and condition (i) of Theorem D.1 trivially holds (since $\mathbb{E}({\boldsymbol{M}}_k \mid {\mathcal F}_{k-1}^{\boldsymbol{X}})={\boldsymbol{0}}$ , $k\in\mathbb{N}$ , and $\boldsymbol{\beta}(t, {\boldsymbol{x}}) = {\boldsymbol{0}}$ , $t\in\mathbb{R}_+$ , ${\boldsymbol{x}}\in\mathbb{R}^3$ ). We now show that conditions (ii) and (iii) of Theorem D.1 hold. We have to check that for all $T \in \mathbb{R}_{++}$

(5.5) \begin{align} \sup\nolimits_{t\in[0,T]} \biggl\Vert\frac{1}{n^2} \sum_{k=1}^{\lfloor nt\rfloor} \mathbb{E}({\boldsymbol{M}}_k {\boldsymbol{M}}_k^{\top} \mid {\mathcal F}_{k-1}^{\boldsymbol{X}}) - \int_0^t \boldsymbol{\mathcal{R}}^{(n)}_s \,{\boldsymbol{V}}_{\boldsymbol{\xi}} \, \mathrm{d} s \biggr\Vert \stackrel{\mathbb{P}}{\longrightarrow} 0 \quad \text{as $n\to\infty$,} \end{align}

(5.6) \begin{align} \frac{1}{n^2} \sum_{k=1}^{\lfloor nT\rfloor} \mathbb{E}(\|{\boldsymbol{M}}_k\|^2 {{\boldsymbol{1}}}_{\{\|{\boldsymbol{M}}_k\|>n\theta\}} \mid {\mathcal F}_{k-1}^{\boldsymbol{X}}) \stackrel{\mathbb{P}}{\longrightarrow} 0 \quad \text{as $n\to\infty$ for all $\theta \in \mathbb{R}_{++}$,} \end{align}

where the process $(\boldsymbol{\mathcal{R}}^{(n)}_s)_{s \in \mathbb{R}_+}$ and the matrix ${\boldsymbol{V}}_{\boldsymbol{\xi}}$ are defined by

\begin{gather*} \boldsymbol{\mathcal{R}}^{(n)}_s \,:\!=\, \begin{bmatrix} ({\mathcal M}_{s,1}^{(n)} + b_1 s)^+ &\quad 0 &\quad 0 \\ 0 &\quad ({\mathcal M}_{s,2}^{(n)} + b_2 s)^+ &\quad 0 \\ 0 &\quad 0 &\quad ({\mathcal M}_{s,3}^{(n)} + b_3 s)^+ \\ \end{bmatrix} , \quad s \in \mathbb{R}_+ , \quad n \in \mathbb{N} , \\ {\boldsymbol{V}}_{\boldsymbol{\xi}} \,:\!=\, \begin{bmatrix} v^{(1)}_{1,1} &\quad 0 &\quad 0 \\ 0 &\quad v^{(2)}_{2,2} &\quad 0 \\ 0 &\quad 0 &\quad v^{(3)}_{3,3} \\ \end{bmatrix}. \end{gather*}

Indeed, $\mathbb{E}( {\boldsymbol{M}}_k\mid {\mathcal F}_{k-1}^{{\boldsymbol{X}}}) = {\boldsymbol{0}}$ , and thus $\text{var}({\boldsymbol{M}}_k\mid{\mathcal F}_{k-1}^{{\boldsymbol{X}}})=\mathbb{E}({\boldsymbol{M}}_k{\boldsymbol{M}}_k^{\top}\mid{\mathcal F}_{k-1}^{{\boldsymbol{X}}})$ , $k\in\mathbb{N}$ , and, since $\gamma(t,{\boldsymbol{x}})$ is symmetric for all $t\in\mathbb{R}_+$ and ${\boldsymbol{x}}\in\mathbb{R}^3$ , we have

\begin{align*} \gamma(t,{\boldsymbol{x}}) \gamma(t,{\boldsymbol{x}})^{\top} & = \begin{bmatrix} \sqrt{v^{(1)}_{1,1}(x_1 + b_1 t)^+} &\quad 0 &\quad 0 \\ 0 &\quad \sqrt{v^{(2)}_{2,2}(x_2 + b_2 t)^+} &\quad 0 \\ 0 &\quad 0 &\quad \sqrt{v^{(3)}_{3,3}(x_3 + b_3 t)^+} \\ \end{bmatrix}^2\\ & = \begin{bmatrix} v^{(1)}_{1,1} (x_1 + b_1 t)^+ &\quad 0 &\quad 0 \\ 0 &\quad v^{(2)}_{2,2}(x_2 + b_2 t)^+ &\quad 0 \\ 0 &\quad 0 & \quad v^{(3)}_{3,3}(x_3 + b_3 t)^+ \end{bmatrix} \\ & = \begin{bmatrix} (x_1 + b_1 t)^+ &\quad 0 &\quad 0 \\ 0 &\quad (x_2 + b_2 t)^+ &\quad 0 \\ 0 &\quad 0 &\quad (x_3 + b_3 t)^+ \end{bmatrix} {\boldsymbol{V}}_{\boldsymbol{\xi}}, \quad t\in\mathbb{R}_+, \quad {\boldsymbol{x}}\in\mathbb{R}^3. \end{align*}

Step 2(b). We now check (5.5). For all $s \in \mathbb{R}_+$ and $n \in \mathbb{N}$ , we have

\[ \boldsymbol{\mathcal{M}}_s^{(n)} + {\boldsymbol{b}} s = \frac{1}{n} {\boldsymbol{X}}_{\lfloor ns\rfloor} + \frac{ns-{\lfloor ns\rfloor}}{n} {\boldsymbol{b}}. \]

Thus,

\[ \boldsymbol{\mathcal{R}}^{(n)}_s = \begin{bmatrix} {\mathcal M}_{s,1}^{(n)} + b_1 s &\quad 0 &\quad 0 \\ 0 &\quad {\mathcal M}_{s,2}^{(n)} + b_2 s &\quad 0 \\ 0 &\quad 0 &\quad {\mathcal M}_{s,3}^{(n)} + b_3 s \end{bmatrix} , \quad s \in \mathbb{R}_+ , \quad n \in \mathbb{N} , \]

and hence

\begin{align*} \int_0^t \boldsymbol{\mathcal{R}}^{(n)}_s \, \mathrm{d} s &= \frac{1}{n^2} \sum_{k=0}^{{\lfloor nt\rfloor}-1} \begin{bmatrix} X_{k,1} &\quad 0 &\quad 0 \\ 0 &\quad X_{k,2} &\quad 0 \\ 0 &\quad 0 &\quad X_{k,3} \\ \end{bmatrix} + \frac{nt-{\lfloor nt\rfloor}}{n^2} \begin{bmatrix} X_{{\lfloor nt\rfloor},1} &\quad 0 &\quad 0 \\ 0 &\quad X_{{\lfloor nt\rfloor},2} &\quad 0 \\ 0 &\quad 0 &\quad X_{{\lfloor nt\rfloor},3} \end{bmatrix} \\ &\quad + \frac{{\lfloor nt\rfloor}+(nt-{\lfloor nt\rfloor})^2}{2n^2} \begin{bmatrix} b_1 &\quad 0 &\quad 0 \\ 0 &\quad b_2 &\quad 0 \\ 0 &\quad 0 &\quad b_3 \\ \end{bmatrix}, \quad t \in \mathbb{R}_+ , \quad n \in \mathbb{N}; \end{align*}

see, for example, the proof of Theorem 1.1 in [Reference Barczy, Bezdány and Pap4]. By Lemma B.1,

\[ \frac{1}{n^2} \sum_{k=1}^{\lfloor nt\rfloor} \mathbb{E}({\boldsymbol{M}}_k {\boldsymbol{M}}_k^{\top} \mid {\mathcal F}_{k-1}^{\boldsymbol{X}}) = \frac{{\lfloor nt\rfloor}}{n^2} {\boldsymbol{V}}^{(0)} + \frac{1}{n^2} \sum_{k=1}^{\lfloor nt\rfloor} \Big(X_{k-1,1} {\boldsymbol{V}}^{(1)} + X_{k-1,2} {\boldsymbol{V}}^{(2)} + X_{k-1,3} {\boldsymbol{V}}^{(3)} \Big) \]

for all $t \in \mathbb{R}_+$ and $n \in \mathbb{N}$ . Since $\xi_{1,1,2,1} \stackrel{{\mathrm{a.s.}}}{=} 0$ , $\xi_{1,1,3,1} \stackrel{{\mathrm{a.s.}}}{=} 0$ , $\xi_{1,1,1,2} \stackrel{{\mathrm{a.s.}}}{=} 0$ , $\xi_{1,1,3,2} \stackrel{{\mathrm{a.s.}}}{=} 0$ , $\xi_{1,1,1,3} \stackrel{{\mathrm{a.s.}}}{=} 0$ , and $\xi_{1,1,2,3} \stackrel{{\mathrm{a.s.}}}{=} 0$ (due to $a_{1,2} = a_{1,3} = a_{2,1} = a_{2,3}= a_{3,1} = a_{3,2} = 0$ ), we have $v^{(1)}_{i,j} = 0$ , $i,j\in\{1,2,3\}$ , $(i,j)\ne (1,1)$ , $v^{(2)}_{i,j} = 0$ , $i,j\in\{1,2,3\}$ , $(i,j)\ne (2,2)$ , and $v^{(3)}_{i,j} = 0$ , $i,j\in\{1,2,3\}$ , $(i,j)\ne (3,3)$ . Consequently, for each $k\in\mathbb{N}$ , we obtain

\begin{align*} &X_{k-1,1} {\boldsymbol{V}}^{(1)} + X_{k-1,2} {\boldsymbol{V}}^{(2)} + X_{k-1,3} {\boldsymbol{V}}^{(3)}\\ &\quad = X_{k-1,1} \begin{bmatrix} v^{(1)}_{1,1} &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 \end{bmatrix} + X_{k-1,2} \begin{bmatrix} 0 &\quad 0 &\quad 0 \\ 0 &\quad v^{(2)}_{2,2} &\quad 0 \\ 0 &\quad 0 &\quad 0 \end{bmatrix} + X_{k-1,3} \begin{bmatrix} 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad v^{(3)}_{3,3} \end{bmatrix}\\ &\quad= \begin{bmatrix} X_{k-1,1} &\quad 0 &\quad 0 \\ 0 &\quad X_{k-1,2} &\quad 0 \\ 0 &\quad 0 &\quad X_{k-1,3} \end{bmatrix} \begin{bmatrix} v^{(1)}_{1,1} &\quad 0 &\quad 0\\ 0 &\quad v^{(2)}_{2,2} &\quad 0 \\ 0 &\quad 0 &\quad v^{(3)}_{3,3} \end{bmatrix} = \begin{bmatrix} X_{k-1,1} &\quad 0 &\quad 0 \\ 0 &\quad X_{k-1,2} &\quad 0 \\ 0 &\quad 0 &\quad X_{k-1,3} \end{bmatrix} {\boldsymbol{V}}_{\boldsymbol{\xi}}. \end{align*}

So

\begin{align*} & \frac{1}{n^2} \sum_{k=1}^{\lfloor nt\rfloor} \mathbb{E}({\boldsymbol{M}}_k {\boldsymbol{M}}_k^{\top} \mid {\mathcal F}_{k-1}^{\boldsymbol{X}}) - \int_0^t \boldsymbol{\mathcal{R}}^{(n)}_s {\boldsymbol{V}}_{\boldsymbol{\xi}} \, \mathrm{d} s \\ &\quad = \frac{{\lfloor nt\rfloor}}{n^2}{\boldsymbol{V}}^{(0)} - \frac{nt-{\lfloor nt\rfloor}}{n^2} \begin{bmatrix} X_{{\lfloor nt\rfloor},1} &\quad 0 &\quad 0 \\ 0 &\quad X_{{\lfloor nt\rfloor},2} &\quad 0 \\ 0 &\quad 0 &\quad X_{{\lfloor nt\rfloor},3} \end{bmatrix} {\boldsymbol{V}}_{\boldsymbol{\xi}}\\ &\quad\, \phantom{=} - \frac{{\lfloor nt\rfloor} + (nt - {\lfloor nt\rfloor})^2}{2n^2} \begin{bmatrix} b_1 &\quad 0 &\quad 0 \\ 0 &\quad b_2 &\quad 0 \\ 0 &\quad 0 &\quad b_3 \end{bmatrix} {\boldsymbol{V}}_{\boldsymbol{\xi}}, \quad t\in\mathbb{R}_+, \quad n\in\mathbb{N}. \end{align*}

Hence, to show (5.5), by Slutsky’s lemma and taking into account the facts that for all $T\in\mathbb{R}_{++}$

\[ \sup\nolimits_{t\in[0,T]} \frac{\lfloor nt\rfloor + (nt - \lfloor nt\rfloor)^2}{n^2} \leq \sup\nolimits_{t\in[0,T]} \frac{\lfloor nt\rfloor + 1}{n^2} \to 0 \quad \text{as $n\to\infty$,} \]

and $\sup_{t\in[0,T]}\frac{\lfloor nt\rfloor}{n^2}{\boldsymbol{V}}^{(0)}\to{\boldsymbol{0}}$ as $n\to\infty$ , it suffices to prove that for all $T\in\mathbb{R}_{++}$ we have

(5.7) \begin{equation} \frac{1}{n^2} \sup\nolimits_{t \in [0,T]} \| (nt-{\lfloor nt\rfloor} ) {\boldsymbol{X}}_{\lfloor nt\rfloor}\| \leq \frac{1}{n^2} \sup\nolimits_{t \in [0,T]} \|{\boldsymbol{X}}_{\lfloor nt\rfloor}\| \stackrel{\mathbb{P}}{\longrightarrow} 0 \quad \text{as $n \to \infty$.} \end{equation}

To prove (5.7), it is enough to show that

\begin{equation*}\mathbb{E}\left( \frac{1}{n^4} \sup\nolimits_{t \in [0,T]} \left \| {\boldsymbol{X}}_{\lfloor nt\rfloor} \right \|^2\right)\to0 \quad \text{as $n\to\infty$.}\end{equation*}

By (4.13), for each $k\in\mathbb{N}$ we have

\begin{equation*} \begin{bmatrix} X_{k,1} \\ X_{k,2} \\ X_{k,3} \end{bmatrix} = \begin{bmatrix} X_{k,1}^{(1)} \\ X_{k,2}^{(2)} \\ X_{k,3}^{(4)} \end{bmatrix} = \begin{bmatrix} \sum_{\ell=1}^k(M_{\ell,1}+b_1) \\ \sum_{\ell=1}^k(M_{\ell,2}+b_2) \\ \sum_{\ell=1}^k(M_{\ell,3}+b_3) \end{bmatrix},\end{equation*}

which together with Lemma B.4 and $\eta_1=\eta_2=\eta_3=1$ (following from Lemma 4.3) yields that

\begin{align*}&\mathbb{E}\left( \frac{1}{n^4} \sup\nolimits_{t \in [0,T]} \left \| {\boldsymbol{X}}_{{\lfloor nt\rfloor}} \right \|^2\right)\\&\quad\quad\leq \frac{1}{n^4}\mathbb{E}\left( \sup\nolimits_{t \in [0,T]} \left(X_{{\lfloor nt\rfloor},1}^{(1)}\right)^2+\sup\nolimits_{t \in [0,T]} \left(X_{{\lfloor nt\rfloor},2}^{(2)}\right)^2+\sup\nolimits_{t \in [0,T]} \left(X_{{\lfloor nt\rfloor},3}^{(4)}\right)^2\right)\\&\quad\quad=\frac{1}{n^4}(\operatorname{O}\!(n^{\eta_1+1})+\operatorname{O}\!(n^{\eta_2+1})+\operatorname{O}\!(n^{\eta_3+1}))=\operatorname{O}\!(n^{-2})\to0\quad \text{as $n\to\infty$,}\end{align*}

implying (5.7), and hence (5.5).

Step 2(c). We check condition (5.6). We show that for all $T\in\mathbb{R}_{++}$ and $\theta\in\mathbb{R}_{++}$ ,

\[ \frac{1}{n^2} \sum_{k=1}^{\lfloor nT\rfloor} \mathbb{E}(\|{\boldsymbol{M}}_k\|^2 {{\boldsymbol{1}}}_{\{\|{\boldsymbol{M}}_k\|>n\theta\}} \mid {\mathcal F}_{k-1}^{\boldsymbol{X}}) \stackrel{{L_1}}{\longrightarrow} 0 \quad \text{as $n\to\infty$.} \]

Using the inequalities ${{\boldsymbol{1}}}_{\{\|{\boldsymbol{M}}_k\|{>}\alpha\}}{<}\frac{\|{\boldsymbol{M}}_k\|^2}{\alpha^2}$ , $k\in\mathbb{N}, \alpha\in\mathbb{R}_{++}$ , and $(a+b+c)^2 \leq 3(a^2+b^2+c^2)$ , $a,b,c\in\mathbb{R}_+$ , and Lemma B.3 (which yields that $\mathbb{E}(M_{k,i}^4) = \operatorname{O}\!(k^2)$ , $k\in\mathbb{N}$ for $i\in\{1,2,3\}$ ), for all $T\in\mathbb{R}_{++}$ and $\theta\in\mathbb{R}_{++}$ , we have

\begin{align*} &\mathbb{E}\left( \frac{1}{n^2} \sum_{k=1}^{\lfloor nT\rfloor} \mathbb{E}(\|{\boldsymbol{M}}_k\|^2 {{\boldsymbol{1}}}_{\{\|{\boldsymbol{M}}_k\|>n\theta\}} \mid {\mathcal F}_{k-1}^{\boldsymbol{X}}) \right) = \frac{1}{n^2} \sum_{k=1}^{\lfloor nT\rfloor} \mathbb{E}(\|{\boldsymbol{M}}_k\|^2 {{\boldsymbol{1}}}_{\{\|{\boldsymbol{M}}_k\|>n\theta\}}) \\ & \leq \frac{1}{n^2} \sum_{k=1}^{\lfloor nT\rfloor} \mathbb{E}\left( \frac{\|{\boldsymbol{M}}_k\|^4}{n^2\theta^2} \right) \leq \frac{3}{n^4 \theta^2} \sum_{k=1}^{\lfloor nT\rfloor} \mathbb{E}(M_{k,1}^4 + M_{k,2}^4 + M_{k,3}^4 ) = \frac{1}{n^4\theta^2} \sum_{k=1}^{\lfloor nT\rfloor} \operatorname{O}\!(k^2) = \operatorname{O}\!(n^{-1})\to 0 \end{align*}

as $n\to\infty$ .

Step 3. Using (5.2) and Lemma C.2, we can prove (3.4). For each $n\in\mathbb{N}$ , by (5.1), we have $(n^{-1}{\boldsymbol{X}}_{{\lfloor nt\rfloor}})_{t\in\mathbb{R}_+} = \Psi^{(n)}(\boldsymbol{\mathcal{M}}^{(n)})$ , where the mapping $\Psi^{(n)} \,:\, \mathsf{D}(\mathbb{R}_+, \mathbb{R}^3) \to \mathsf{D}(\mathbb{R}_+, \mathbb{R}^3)$ is given by

\[ (\Psi^{(n)}(f))(t) \,:\!=\, f\biggl(\frac{{\lfloor nt\rfloor}}{n}\biggr) + \frac{{\lfloor nt\rfloor}}{n} {\boldsymbol{b}} \]

for $f \in \mathsf{D}(\mathbb{R}_+, \mathbb{R}^3)$ and $t \in \mathbb{R}_+$ . Further, using that $(\boldsymbol{\mathcal{M}}_t +{\boldsymbol{b}} t)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{=} (\boldsymbol{\mathcal{X}}_t)_{t\in\mathbb{R}_+}$ , we have $\boldsymbol{\mathcal{X}} \stackrel{\mathrm{D}}{=} \Psi(\boldsymbol{\mathcal{M}})$ , where the mapping $\Psi \,:\, \mathsf{D}(\mathbb{R}_+, \mathbb{R}^3) \to \mathsf{D}(\mathbb{R}_+, \mathbb{R}^3)$ is given by

\[ (\Psi(f))(t) \,:\!=\, f(t) + {\boldsymbol{b}} t , \quad f \in \mathsf{D}(\mathbb{R}_+, \mathbb{R}^3) , \quad t \in \mathbb{R}_+. \]

The mappings $\Psi^{(n)}$ , $n\in\mathbb{N}$ , and $\Psi$ are measurable, which can be checked in the same way as in step 4(a) in [Reference Barczy, Bezdány and Pap4] by replacing $\mathsf{D}(\mathbb{R}_+,\mathbb{R})$ by $\mathsf{D}(\mathbb{R}_+,\mathbb{R}^3)$ in the argument given there. It can also be checked that the set $C \,:\!=\, \mathsf{C}(\mathbb{R}_+, \mathbb{R}^3)$ satisfies $C \in {\mathcal B}(\mathsf{D}(\mathbb{R}_+, \mathbb{R}^3))$ , $\mathbb{P}(\boldsymbol{\mathcal{M}} \in C) = 1$ , and $\Psi^{(n)}(f^{(n)}) \to \Psi(f)$ in $\mathsf{D}(\mathbb{R}_+, \mathbb{R}^3)$ as $n \to \infty$ if $f^{(n)} \to f$ in $\mathsf{D}(\mathbb{R}_+, \mathbb{R}^3)$ as $n \to \infty$ with $f \in C$ , $f^{(n)}\in \mathsf{D}(\mathbb{R}_+, \mathbb{R}^3)$ , $n\in\mathbb{N}$ . Namely, we can follow the same argument as in step 4(b) in [Reference Barczy, Bezdány and Pap4] by replacing $\mathsf{D}(\mathbb{R}_+,\mathbb{R})$ by $\mathsf{D}(\mathbb{R}_+,\mathbb{R}^3)$ , and $\mathsf{C}(\mathbb{R}_+,\mathbb{R})$ by $\mathsf{C}(\mathbb{R}_+,\mathbb{R}^3)$ , respectively, in the argument given there. So we can apply Lemma C.2, and we obtain $(n^{-1} {\boldsymbol{X}}_{\lfloor nt\rfloor})_{t\in\mathbb{R}_+} = \Psi^{(n)}(\boldsymbol{\mathcal{M}}^{(n)}) \stackrel{\mathrm{D}}{\longrightarrow} \Psi(\boldsymbol{\mathcal{M}})$ as $n \to \infty$ , where $( (\Psi(\boldsymbol{\mathcal{M}}))(t))_{t\in\mathbb{R}_+} = (\boldsymbol{\mathcal{M}}_t + {\boldsymbol{b}} t)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{=} (\boldsymbol{\mathcal{X}}_t)_{t\in\mathbb{R}_+}$ , as desired.

6. Proof of Theorem 3.2

By (4.13), we have the decomposition

(6.1) \begin{align} \begin{bmatrix} X_{{\lfloor nt\rfloor},1} \\ X_{{\lfloor nt\rfloor},2} \\ X_{{\lfloor nt\rfloor},3} \end{bmatrix} = \begin{bmatrix} X_{{\lfloor nt\rfloor},1}^{(1)} \\[1mm] X^{(2)}_{{\lfloor nt\rfloor},2} \\[1mm] a_{3,1}X_{{\lfloor nt\rfloor},3}^{(2)}+a_{3,2}X_{{\lfloor nt\rfloor},3}^{(3)}+X_{{\lfloor nt\rfloor},3}^{(4)} \end{bmatrix} , \quad t \in \mathbb{R}_+ , \quad n \in \mathbb{N}, \end{align}

where

\begin{gather*} X^{(1)}_{{\lfloor nt\rfloor},1} = \sum_{j=1}^{\lfloor nt\rfloor} (M_{j,1} + b_1), \quad X^{(2)}_{{\lfloor nt\rfloor},2} = \sum_{j=1}^{\lfloor nt\rfloor} (M_{j,2} + b_2), \quad X_{{\lfloor nt\rfloor},3}^{(4)} = \sum_{j=1}^{\lfloor nt\rfloor} (M_{j,3} + b_3),\end{gather*}

and due to $X_{{\lfloor nt\rfloor},1}=X_{{\lfloor nt\rfloor},1}^{(1)}$ , $X_{{\lfloor nt\rfloor},2}=X_{{\lfloor nt\rfloor},2}^{(2)}$ and (4.14),

\begin{gather*} X^{(2)}_{{\lfloor nt\rfloor},3}=n^2\int_0^{ \frac{{\lfloor nt\rfloor}}{n}} n^{-1}X_{{\lfloor ns\rfloor},1} \, \mathrm{d} s, \quad X^{(3)}_{{\lfloor nt\rfloor},3} =n^2\int_0^{ \frac{{\lfloor nt\rfloor}}{n}} n^{-1}X_{{\lfloor ns\rfloor},2} \, \mathrm{d} s \end{gather*}

for $t\in\mathbb{R}_+$ and $n\in\mathbb{N}$ .

Step 1. First, we show that

(6.2) \begin{equation} \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-1} X_{{\lfloor nt\rfloor},2} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{equation}

where $([{\mathcal X}_{t,1},{\mathcal X}_{t,2}]^{\top})_{t\in\mathbb{R}_+}$ is the pathwise unique strong solution of the first two equations of the SDE (3.6) with initial value $[{\mathcal X}_{t,1},{\mathcal X}_{t,2}]^{\top}={\boldsymbol{0}}\in\mathbb{R}^2$ . Let us define

\[{\boldsymbol{R}}\,:\!=\, \begin{bmatrix} 1 &\quad 0 &\quad 0\\ 0 &\quad 1 &\quad 0 \end{bmatrix},\]

and consider the stochastic process $({\boldsymbol{Y}}_k)_{k\in\mathbb{Z}_+}\,:\!=\,([Y_{k,1},Y_{k,2}]^{\top})_{k\in\mathbb{Z}_+}\,:\!=\,({\boldsymbol{R}}{\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ . It is easy to see that $({\boldsymbol{Y}}_k)_{k\in\mathbb{Z}_+}$ is a 2-type GWI process with initial value ${\boldsymbol{Y}}_0={\boldsymbol{0}}\in\mathbb{R}^2$ , offspring distributions ${\boldsymbol{R}}\boldsymbol{\xi}_{i}$ , $i\in\{1,2\}$ , and immigration distribution ${\boldsymbol{R}}{\boldsymbol{\varepsilon}}$ . Indeed, we have ${\boldsymbol{Y}}_0={\boldsymbol{R}}{\boldsymbol{X}}_0={\boldsymbol{0}}\in\mathbb{R}^2$ , and since $Y_{k,i}=({\boldsymbol{R}}{\boldsymbol{X}}_k)_i=X_{k,i}$ , $i\in\{1,2\}$ , by (3.1), we get

\begin{align*} {\boldsymbol{Y}}_k={\boldsymbol{R}}{\boldsymbol{X}}_k =& \sum_{j=1}^{X_{k-1,1}} {\boldsymbol{R}}\boldsymbol{\xi}_{k,j,1} + \sum_{j=1}^{X_{k-1,2}} {\boldsymbol{R}}\boldsymbol{\xi}_{k,j,2} + \sum_{j=1}^{X_{k-1,3}} {\boldsymbol{R}}\boldsymbol{\xi}_{k,j,3} + {\boldsymbol{R}}{\boldsymbol{\varepsilon}}_{k}\\ =& \sum_{j=1}^{Y_{k-1,1}} {\boldsymbol{R}}\boldsymbol{\xi}_{k,j,1} + \sum_{j=1}^{Y_{k-1,2}} {\boldsymbol{R}}\boldsymbol{\xi}_{k,j,2} + {\boldsymbol{R}}{\boldsymbol{\varepsilon}}_{k}, \quad k \in \mathbb{N}, \end{align*}

where we used that ${\boldsymbol{R}}\boldsymbol{\xi}_3={\boldsymbol{0}}\in\mathbb{R}^2$ . Note that the conditions of Theorem 3.2 imply that $\mathbb{E}(\|{\boldsymbol{R}}\boldsymbol{\xi}_i\|^4)<\infty$ , $i\in\{1,2\}$ , and $\mathbb{E}(\|{\boldsymbol{R}}{\boldsymbol{\varepsilon}}\|^4)<\infty$ . Furthermore, the offspring mean matrix of the GWI process $({\boldsymbol{Y}}_k)_{k\in\mathbb{Z}_+}$ is ${\boldsymbol{I}}_2$ (since the offspring mean matrix $({\boldsymbol{X}}_k)_{k\in\mathbb{Z}_+}$ satisfies (2) of (3.2)), and $\mathbb{E}({\boldsymbol{R}}{\boldsymbol{\varepsilon}})={\boldsymbol{R}}{\boldsymbol{b}}=[b_1,b_2]^{\top}$ . Therefore, $({\boldsymbol{Y}}_k)_{k\in\mathbb{Z}_+}$ satisfies all conditions of Theorem 2.1 in [Reference Barczy, Bezdány and Pap5], and consequently we have

\begin{equation*} \left(n^{-1}{\boldsymbol{Y}}_{\lfloor nt\rfloor}\right)_{t\in\mathbb{R}_+} =\left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-1} X_{{\lfloor nt\rfloor},2} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$,} \end{equation*}

as desired. For a direct proof of (6.2) (similar to the proof of Theorem 3.1), see the arXiv version of this paper [Reference Barczy and Bezdány3].

Step 2(a). Next, we turn to proving the main assertion. By (4.14), we have

(6.3) \begin{align} n^{-2} X^{(2)}_{{\lfloor nt\rfloor},3}=\int_0^{ \frac{{\lfloor nt\rfloor}}{n}} n^{-1}X_{{\lfloor ns\rfloor},1}^{(1)} \, \mathrm{d} s, \quad n^{-2} X^{(3)}_{{\lfloor nt\rfloor},3}=\int_0^{ \frac{{\lfloor nt\rfloor}}{n}} n^{-1}X_{{\lfloor ns\rfloor},2}^{(2)} \, \mathrm{d} s \end{align}

for all $t \in \mathbb{R}_+$ and $n \in \mathbb{N}$ . Using (6.2) and Lemmas C.3 and C.8, we check that

(6.4) \begin{align} \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-1} X_{{\lfloor nt\rfloor},2} \\ a_{3,1}n^{-2} X^{(2)}_{{\lfloor nt\rfloor},3} + a_{3,2}n^{-2} X^{(3)}_{{\lfloor nt\rfloor},3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2} \\ \int_0^t(a_{3,1}{\mathcal X}_{s,1}+a_{3,2}{\mathcal X}_{s,2}) \, \mathrm{d} s \end{bmatrix}\right)_{t\in\mathbb{R}_+} \end{align}

as $n \to \infty$ . Namely, let us apply Lemma C.8 with $d=2$ , and the functions $\Phi$ and $\Phi_n$ , $n\in\mathbb{N}$ , given there, and Lemma C.3 with

\begin{equation*}{\boldsymbol{B}}\,:\!=\,\begin{bmatrix}1 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0\\0 &\quad 1 &\quad 0 &\quad 0 &\quad 0 &\quad 0\\0 &\quad 0 &\quad a_{3,1} &\quad a_{3,2} &\quad 0 &\quad 0\\\end{bmatrix}.\end{equation*}

Then, by (6.3), we have

\begin{align*} \begin{bmatrix} {\mathcal X}_{t,1} \\[4pt] {\mathcal X}_{t,2} \\[4pt] a_{3,1}\int_0^t{\mathcal X}_{s,1}\, \mathrm{d} s+a_{3,2}\int_0^t{\mathcal X}_{s,2} \, \mathrm{d} s \end{bmatrix} &= {\boldsymbol{B}} \left(\Phi\left( \left(\begin{bmatrix} {\mathcal X}_{u,1}\\[4pt] {\mathcal X}_{u,2} \end{bmatrix}\right)_{u\in\mathbb{R}_+}\right)\right)(t),\\[4pt] \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\[4pt] n^{-1} X_{{\lfloor nt\rfloor},2} \\[4pt] a_{3,1}n^{-2} X^{(2)}_{{\lfloor nt\rfloor},3} + a_{3,2}n^{-2} X^{(3)}_{{\lfloor nt\rfloor},3} \end{bmatrix} &={\boldsymbol{B}} \left(\Phi_n\left( \left(n^{-1} \begin{bmatrix} X_{\lfloor nu\rfloor,1}\\[4pt] X_{\lfloor nu\rfloor,2} \end{bmatrix}\right)_{u\in\mathbb{R}_+}\right)\right)(t)\end{align*}

for $t\in\mathbb{R}_+$ and $n\in\mathbb{N}$ . Since $([{\mathcal X}_{u,1},{\mathcal X}_{u,2}]^{\top})_{u\in\mathbb{R}_+}$ has continuous sample paths almost surely, (6.2) and Lemma C.8 yield that

\[ \Phi_n\left( \left(n^{-1} \begin{bmatrix} X_{\lfloor nu\rfloor,1}\\ X_{\lfloor nu\rfloor,2} \end{bmatrix}\right)_{u\in\mathbb{R}_+}\right) \stackrel{\mathrm{D}}{\longrightarrow} \Phi\left( \left(\begin{bmatrix} {\mathcal X}_{u,1}\\ {\mathcal X}_{u,2} \end{bmatrix}\right)_{u\in\mathbb{R}_+}\right) \quad \text{as $n\to\infty$.} \]

An application of Lemma C.3 then implies (6.4), as desired.

Step 2(b). Next we show that

(6.5) \begin{equation}\sup\nolimits_{t\in[0,T]}\left\Vert\begin{bmatrix}0\\0\\n^{-2}X_{{\lfloor nt\rfloor},3}^{(4)}\end{bmatrix}\right\Vert\stackrel{\mathbb{P}}{\longrightarrow}0\quad\text{as $n \to \infty$ for all $T \in \mathbb{R}_{++}$.}\end{equation}

Note that for all $T\in\mathbb{R}_+$ and $n\in\mathbb{N}$ , the left-hand side of (6.5) is equal to $n^{-2}\sup_{t\in[0,T]}\left|X_{{\lfloor nt\rfloor},3}^{(4)}\right|$ . By (4.13) and Lemma B.4, for all $T\in\mathbb{R}_+$ we have

\begin{align*} \mathbb{E}\left(n^{-4}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|^2\right) &=n^{-4}\mathbb{E}\left(\sup\nolimits_{t\in[0,T]}\left(\sum_{j=1}^{{\lfloor nt\rfloor}}(M_{j,3}+b_3)\right)^2 \right)\\ &=\operatorname{O}\!(n^{-4+\eta_3+1}),\quad n\in\mathbb{N}.\end{align*}

By Lemma 4.3, we have $\eta_3=2$ , and hence $\operatorname{O}\!(n^{-4+ \eta_3+1})\to 0$ as $n\to\infty$ , yielding that $\left(n^{-2}\sup_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|\right)^2\stackrel{\mathbb{P}}{\longrightarrow}0$ as $n\to\infty$ for all $T\in\mathbb{R}_+$ . This implies (6.5).

Step 2(c). Consequently, by Lemma C.9 (a kind of Slutsky’s lemma for stochastic processes with trajectories in $\mathsf{D}(\mathbb{R}_+, \mathbb{R}^d)$ ), the decomposition (6.1) and the convergences (6.4) and (6.5) yield that

\[ \left( \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-1} X_{{\lfloor nt\rfloor},2} \\ n^{-2} X_{{\lfloor nt\rfloor},3} \end{bmatrix} \right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ {\mathcal X}_{t,2} \\ \int_0^t a_{3,1}{\mathcal X}_{s,1}+a_{3,2}{\mathcal X}_{s,2} \, \mathrm{d} s \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$.} \]

By Itô’s formula, the limit process above is the pathwise unique strong solution of the SDE (3.6) with initial value $[{\mathcal X}_{0,1},{\mathcal X}_{0,2},{\mathcal X}_{0,3}]^{\top}={\boldsymbol{0}}$ , and thus we get the assertion of Theorem 3.2.

7. Proof of Theorem 3.3

By (4.13), we have the decomposition

(7.1) \begin{align} \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-2} X_{{\lfloor nt\rfloor},2} \\ n^{-2} X_{{\lfloor nt\rfloor},3} \end{bmatrix} = \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1}^{(1)} \\[1mm] a_{2,1} \, n^{-2} X^{(1)}_{{\lfloor nt\rfloor},2} + n^{-2} X^{(2)}_{{\lfloor nt\rfloor},2} \\[1mm] a_{3,1} \, n^{-2} X_{{\lfloor nt\rfloor},3}^{(2)} + n^{-2} X_{{\lfloor nt\rfloor},3}^{(4)} \end{bmatrix} , \quad t \in \mathbb{R}_+ , \quad n \in \mathbb{N} , \end{align}

where

\begin{equation*} X^{(2)}_{{\lfloor nt\rfloor},2} = \sum_{j=1}^{\lfloor nt\rfloor} (M_{j,2} + b_2), \quad X_{{\lfloor nt\rfloor},3}^{(4)} = \sum_{j=1}^{{\lfloor nt\rfloor}} (M_{j,3} + b_3), \end{equation*}

and, by (4.14),

(7.2) \begin{equation} X^{(1)}_{{\lfloor nt\rfloor},2} = X^{(2)}_{{\lfloor nt\rfloor},3} = n^2 \int_0^{{{\lfloor nt\rfloor}}/{n}} n^{-1}X_{{\lfloor ns\rfloor},1}^{(1)} \, \mathrm{d} s, \quad t\in\mathbb{R}_+, \quad n\in\mathbb{N}. \end{equation}

Step 1. By (3.3), we have

(7.3) \begin{equation}(n^{-1} X_{{\lfloor nt\rfloor},1})_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} ({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}\quad \text{as $n \to \infty$,}\end{equation}

where $({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ is the pathwise unique strong solution of the first equation of the SDE (3.7) with ${\mathcal X}_{0,1}=0$ .

Step 2(a). We show that

(7.4) \begin{equation} \begin{split} &\left(\begin{bmatrix} n^{-1}X_{{\lfloor nt\rfloor},1}\\ a_{2,1}n^{-2} X_{{\lfloor nt\rfloor},2}^{(2)}\\ a_{3,1}n^{-2} X_{{\lfloor nt\rfloor},3}^{(4)} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ a_{2,1} \int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \\ a_{3,1}\int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{ as $n\to\infty$.}\end{split}\end{equation}

Let us apply Lemma C.8 with $d=1$ and the functions $\Phi$ and $\Phi_n$ , $n\in\mathbb{N}$ , given there, and Lemma C.3 with

\begin{equation*}{\boldsymbol{B}}\,:\!=\,\begin{bmatrix}1 &\quad 0 &\quad 0 \\0 &\quad a_{2,1} &\quad 0 \\0 &\quad a_{3,1} &\quad 0\end{bmatrix}.\end{equation*}

Then, by (7.2), we have

\begin{equation*} \begin{bmatrix} {\mathcal X}_{t,1} \\ a_{2,1}\int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \\ a_{3,1}\int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \end{bmatrix} \! = {\boldsymbol{B}} \!\left(\Phi(({\mathcal X}_{u,1})_{u\in\mathbb{R}_+})\right)(t), \quad \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ a_{2,1} n^{-2} X^{(1)}_{{\lfloor nt\rfloor},2} \\ a_{3,1} n^{-2} X^{(1)}_{{\lfloor nt\rfloor},2} \end{bmatrix}\! = {\boldsymbol{B}}(\Phi_n( (n^{-1}X_{\lfloor nu \rfloor,1})_{u\in\mathbb{R}_+} ))(t)\end{equation*}

for $t\in\mathbb{R}_+$ and $n\in\mathbb{N}$ . Since $({\mathcal X}_{u,1})_{u\in\mathbb{R}_+}$ has continuous sample paths almost surely, Lemma C.8 and (7.3) imply that $\Phi_n( (n^{-1}X_{\lfloor nu\rfloor,1})_{u\in\mathbb{R}_+} )\stackrel{\mathrm{D}}{\longrightarrow} \Phi( ({\mathcal X}_{u,1})_{u\in\mathbb{R}_+} )$ as $n\to\infty$ . Then an application of Lemma C.3 yields (7.4), as desired.

Step 2(b). The aim of the following discussion is to show that

(7.5) \begin{equation}\sup\nolimits_{t\in[0,T]}\left\Vert\begin{bmatrix}0\\n^{-2}X_{{\lfloor nt\rfloor},2}^{(2)}\\[4pt]n^{-2}X_{{\lfloor nt\rfloor},3}^{(4)}\end{bmatrix}\right\Vert\stackrel{\mathbb{P}}{\longrightarrow}0\quad\text{as $n \to \infty$ for all $T \in \mathbb{R}_{++}$.}\end{equation}

Note that for all $T\in\mathbb{R}_+$ and $n\in\mathbb{N}$ , we have

\begin{equation*}\sup\nolimits_{t\in[0,T]}\left\Vert\begin{bmatrix}0\\n^{-2}X_{{\lfloor nt\rfloor},2}^{(2)}\\[4pt]n^{-2}X_{{\lfloor nt\rfloor},3}^{(4)}\end{bmatrix}\right\Vert\leq n^{-2}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},2}^{(2)}| +n^{-2}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|.\end{equation*}

By (4.13) and Lemma B.4, for all $T\in\mathbb{R}_+$ we have

\begin{align*}\mathbb{E}\left(n^{-4}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},2}^{(2)}|^2\right) &=n^{-4}\mathbb{E}\left(\sup\nolimits_{t\in[0,T]}\left(\sum_{j=1}^{{\lfloor nt\rfloor}}(M_{j,2}+b_2)\right)^2 \right)\\ &=\operatorname{O}\!(n^{-4+\eta_2+1}),\quad n\in\mathbb{N},\\\mathbb{E}\left(n^{-4}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|^2\right) &=n^{-4}\mathbb{E}\left(\sup\nolimits_{t\in[0,T]}\left(\sum_{j=1}^{{\lfloor nt\rfloor}}(M_{j,3}+b_3)\right)^2 \right)\\ &=\operatorname{O}\!(n^{-4+\eta_3+1}),\quad n\in\mathbb{N}.\end{align*}

By Lemma 4.3, we have $\eta_2=\eta_3=2$ , and hence $\operatorname{O}\!(n^{-4+\eta_i+1})\to 0$ as $n\to\infty$ , $i\in\{2,3\}$ , yielding that

\begin{gather*}\left(n^{-2}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},2}^{(2)}|\right)^2\stackrel{\mathbb{P}}{\longrightarrow}0\quad \text{as $n\to\infty$ for all $T\in\mathbb{R}_+$,}\\\left(n^{-2}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|\right)^2\stackrel{\mathbb{P}}{\longrightarrow}0\quad \text{as $n\to\infty$ for all $T\in\mathbb{R}_+$.}\end{gather*}

This implies (7.5).

Step 2(c). Consequently, by Lemma C.9, (a kind of Slutsky’s lemma for stochastic processes with trajectories in $\mathsf{D}(\mathbb{R}_+, \mathbb{R}^d)$ ), the decomposition (7.1) and the convergences (7.4) and (7.5) yield that

\[ \left(\begin{bmatrix}n^{-1} X_{{\lfloor nt\rfloor},1}\\n^{-2} X_{{\lfloor nt\rfloor},2}\\n^{-2} X_{{\lfloor nt\rfloor},3}\end{bmatrix}\right)_{t\in\mathbb{R}_+}\stackrel{\mathrm{D}}{\longrightarrow}\left(\begin{bmatrix} {\mathcal X}_{t,1} \\ a_{2,1} \int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \\[1mm] a_{3,1} \int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \end{bmatrix}\right)_{t\in\mathbb{R}_+}\quad\text{as $n \to \infty$.} \]

By Itô’s formula, the limit process above is the pathwise unique strong solution of the SDE (3.7) with initial value $[{\mathcal X}_{0,1},{\mathcal X}_{0,2},{\mathcal X}_{0,3}]^{\top} = {\boldsymbol{0}}$ , and thus we get the statement of Theorem 3.3.

8. Proof of Theorem 3.4

By (4.13), we have the decomposition

(8.1) \begin{align} \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-2} X_{{\lfloor nt\rfloor},2} \\ n^{-3} X_{{\lfloor nt\rfloor},3} \end{bmatrix} = \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\[1mm] a_{2,1} \, n^{-2} X^{(1)}_{{\lfloor nt\rfloor},2} + n^{-2} X^{(2)}_{{\lfloor nt\rfloor},2} \\[1mm] a_{3,2}a_{2,1}\, n^{-3} X_{{\lfloor nt\rfloor},3}^{(1)} + a_{3,1} \, n^{-3} X_{{\lfloor nt\rfloor},3}^{(2)} + a_{3,2} \, n^{-3} X_{{\lfloor nt\rfloor},3}^{(3)} + n^{-3} X_{{\lfloor nt\rfloor},3}^{(4)} \end{bmatrix} \end{align}

for all $t\in\mathbb{R}_+$ and $n\in\mathbb{N}$ , where

\begin{gather*}X^{(2)}_{{\lfloor nt\rfloor},2} = \sum_{j=1}^{\lfloor nt\rfloor} (M_{j,2} + b_2), \quad X_{{\lfloor nt\rfloor},3}^{(4)} = \sum_{j=1}^{{\lfloor nt\rfloor}} (M_{j,3} + b_3),\\X_{{\lfloor nt\rfloor},3}^{(2)} = \sum_{\ell=1}^{\lfloor nt\rfloor} ({\lfloor nt\rfloor}-\ell) (M_{\ell,1} + b_1),\quad X_{{\lfloor nt\rfloor},3}^{(3)} = \sum_{\ell=1}^{\lfloor nt\rfloor} ({\lfloor nt\rfloor}-\ell) (M_{\ell,2} + b_2),\end{gather*}

and, by (4.14),

(8.2) \begin{gather}X^{(1)}_{{\lfloor nt\rfloor},2} = {n^2}\int_0^{{{\lfloor nt\rfloor}}/{n}} n^{-1} X_{{\lfloor ns\rfloor},1}^{(1)} \, \mathrm{d} s,\quad X_{{\lfloor nt\rfloor},3}^{(1)}= n^3\int_0^{{{\lfloor nt\rfloor}}/{n}}\left(\int_0^{{{\lfloor nr\rfloor}}/{n}} n^{-1} X_{{\lfloor ns\rfloor},1}^{(1)}\,\mathrm{d} s\right)\,\mathrm{d} r.\end{gather}

Step 1. By (3.3), we have

(8.3) \begin{equation} (n^{-1} X_{{\lfloor nt\rfloor},1})_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} ({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}\quad \text{as $n \to \infty$,} \end{equation}

where $({\mathcal X}_{t,1})_{t\in\mathbb{R}_+}$ is the pathwise unique strong solution of the first equation of the SDE (3.8) with ${\mathcal X}_{0,1}=0$ .

Step 2(a) We show that

(8.4) \begin{align} \begin{split} \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ a_{2,1} n^{-2} X^{(1)}_{{\lfloor nt\rfloor},2} \\[4pt] a_{3,2}a_{2,1} n^{-3} X^{(1)}_{{\lfloor nt\rfloor},3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow} \left(\begin{bmatrix} {\mathcal X}_{t,1} \\ a_{2,1}\int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \\[4pt] a_{3,2}a_{2,1}\int_0^t\left(\int_0^r {\mathcal X}_{s,1} \, \mathrm{d} s\right)\,\mathrm{d} r \end{bmatrix}\right)_{t\in\mathbb{R}_+} \end{split} \end{align}

as $n \to \infty$ . Let us apply Lemma C.8 with $d=1$ and the functions $\Phi$ and $\Phi_n$ , $n\in\mathbb{N}$ , given there, and Lemma C.3 with

\begin{equation*}{\boldsymbol{B}}\,:\!=\,\begin{bmatrix}1 &\quad 0 &\quad 0 \\0 &\quad a_{2,1} &\quad 0 \\0 &\quad 0 &\quad a_{3,2}a_{2,1}\end{bmatrix}.\end{equation*}

Then, by (8.2), for all $t\in\mathbb{R}_+$ , we have

\begin{align*} \begin{bmatrix} {\mathcal X}_{t,1} \\ a_{2,1}\int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \\[4pt] a_{3,2}a_{2,1}\int_0^t( \int_0^r{\mathcal X}_{s,1} \, \mathrm{d} s)\,\mathrm{d} r \end{bmatrix} & = {\boldsymbol{B}}\left(\Phi(({\mathcal X}_{u,1})_{u\in\mathbb{R}_+})\right)(t),\\[1mm] \begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ a_{2,1}n^{-2} X^{(1)}_{{\lfloor nt\rfloor},2} \\[4pt] a_{3,2}a_{2,1}n^{-3} X^{(1)}_{{\lfloor nt\rfloor},3} \end{bmatrix} & = {\boldsymbol{B}} \left(\Phi_n( (n^{-1}X_{\lfloor nu \rfloor,1})_{u\in\mathbb{R}_+}) \right)(t), \quad n\in\mathbb{N}.\end{align*}

Since $({\mathcal X}_{u,1})_{u\in\mathbb{R}_+}$ has continuous sample paths almost surely, Lemma C.8 and (8.3) yield that $\Phi_n( (n^{-1}X_{\lfloor nu\rfloor,1})_{u\in\mathbb{R}_+} )\stackrel{\mathrm{D}}{\longrightarrow} \Phi(({\mathcal X}_{u,1})_{u\in\mathbb{R}_+})$ as $n\to\infty$ . Then an application of Lemma C.3 implies (8.4), as desired.

Step 2(b). The aim of the following discussion is to show that

(8.5) \begin{equation}\sup\nolimits_{t\in[0,T]}\left\Vert\begin{bmatrix}n^{-1}(X_{{\lfloor nt\rfloor},1}-X_{{\lfloor nt\rfloor},1}^{(1)})\\[4pt]n^{-2}(X_{{\lfloor nt\rfloor},2}-a_{2,1}X_{{\lfloor nt\rfloor},2}^{(1)})\\[4pt]n^{-3}(X_{{\lfloor nt\rfloor},3}-a_{3,2}a_{2,1}X_{{\lfloor nt\rfloor},3}^{(1)})\end{bmatrix}\right\Vert\stackrel{\mathbb{P}}{\longrightarrow}0\quad \text{as $n \to\infty$ for all $T\in\mathbb{R}_{++}$.}\end{equation}

Note that for all $T\in\mathbb{R}_+$ and $n\in\mathbb{N}$ we have

\begin{align*}&\sup\nolimits_{t\in[0,T]}\left\Vert\begin{bmatrix}n^{-1}(X_{{\lfloor nt\rfloor},1}-X_{{\lfloor nt\rfloor},1}^{(1)})\\[4pt]n^{-2}(X_{{\lfloor nt\rfloor},2}-a_{2,1}X_{{\lfloor nt\rfloor},2}^{(1)})\\[4pt]n^{-3}(X_{{\lfloor nt\rfloor},3}-a_{3,2}a_{2,1}X_{{\lfloor nt\rfloor},3}^{(1)})\end{bmatrix}\right\Vert\\&\quad\quad=\sup\nolimits_{t\in[0,T]}\left\Vert\begin{bmatrix}0\\n^{-2}X_{{\lfloor nt\rfloor},2}^{(2)}\\a_{3,1} \, n^{-3} X_{{\lfloor nt\rfloor},3}^{(2)} + a_{3,2} \, n^{-3} X_{{\lfloor nt\rfloor},3}^{(3)} + n^{-3} X_{{\lfloor nt\rfloor},3}^{(4)}\end{bmatrix}\right\Vert\\&\quad\quad\leq n^{-2}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},2}^{(2)}| +a_{3,1}n^{-3}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(2)}|\\&\quad\quad\phantom{=}\,+a_{3,2}n^{-3}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(3)}| +n^{-3}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|.\end{align*}

By (4.13) and Lemma B.4, for all $T\in\mathbb{R}_+$ we have

\begin{align*}\mathbb{E}\left(n^{-4}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},2}^{(2)}|^2\right) &=n^{-4}\mathbb{E}\left(\sup\nolimits_{t\in[0,T]}\left(\sum_{j=1}^{{\lfloor nt\rfloor}}(M_{j,2}+b_2)\right)^2 \right)\\ &=\operatorname{O}\!(n^{-4+\eta_2+1}), \quad n\in\mathbb{N},\\\mathbb{E}\left(n^{-6}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(2)}|^2\right) &=n^{-6}\mathbb{E}\left(\sup\nolimits_{t\in[0,T]}\left(\sum_{j=1}^{{\lfloor nt\rfloor}}({\lfloor nt\rfloor}-j)(M_{j,1}+b_1)\right)^2 \right)\\ &=\operatorname{O}\!(n^{-6+\eta_1+3}), \quad n\in\mathbb{N},\\\mathbb{E}\left(n^{-6}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(3)}|^2\right) &=n^{-6}\mathbb{E}\left(\sup\nolimits_{t\in[0,T]}\left(\sum_{j=1}^{{\lfloor nt\rfloor}}({\lfloor nt\rfloor}-j)(M_{j,2}+b_2)\right)^2 \right)\\ &=\operatorname{O}\!(n^{-6+\eta_2+3}), \quad n\in\mathbb{N},\\\mathbb{E}\left(n^{-6}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|^2\right) &=n^{-6}\mathbb{E}\left(\sup\nolimits_{t\in[0,T]}\left(\sum_{j=1}^{{\lfloor nt\rfloor}}(M_{j,3}+b_3)\right)^2 \right)\\ &=\operatorname{O}\!(n^{-6+\eta_3+1}), \quad n\in\mathbb{N}.\end{align*}

By Lemma 4.3, we have $\eta_1=1$ , $\eta_2=2$ , and $\eta_3=3$ , and hence $\operatorname{O}\!(n^{-4+\eta_2+1})\to 0$ as $n\to\infty$ , $\operatorname{O}\!(n^{-6+\eta_i+3})\to 0$ as $n\to\infty$ , $i\in\{1,2\}$ , and $\operatorname{O}\!(n^{-6+\eta_3+1})\to 0$ as $n\to\infty$ , yielding that

\begin{gather*}\left(n^{-2}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},2}^{(2)}|\right)^2\stackrel{\mathbb{P}}{\longrightarrow}0\quad \text{as $n\to\infty$ for all $T\in\mathbb{R}_+$,}\\\left(n^{-3}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(2)}|\right)^2\stackrel{\mathbb{P}}{\longrightarrow}0\quad \text{as $n\to\infty$ for all $T\in\mathbb{R}_+$,}\\\left(n^{-3}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(3)}|\right)^2\stackrel{\mathbb{P}}{\longrightarrow}0\quad \text{as $n\to\infty$ for all $T\in\mathbb{R}_+$,}\\\left(n^{-3}\sup\nolimits_{t\in[0,T]}|X_{{\lfloor nt\rfloor},3}^{(4)}|\right)^2\stackrel{\mathbb{P}}{\longrightarrow}0\quad \text{as $n\to\infty$ for all $T\in\mathbb{R}_+$.}\end{gather*}

This implies (8.5).

Step 2(c). Consequently, by Lemma C.9 (a kind of Slutsky’s lemma for stochastic processes with trajectories in $\mathsf{D}(\mathbb{R}_+, \mathbb{R}^d)$ ), the decomposition (8.1) and the convergences (8.4) and (8.5) yield that

\[ \left(\begin{bmatrix} n^{-1} X_{{\lfloor nt\rfloor},1} \\ n^{-2} X_{{\lfloor nt\rfloor},2} \\ n^{-3} X_{{\lfloor nt\rfloor},3} \end{bmatrix}\right)_{t\in\mathbb{R}_+} \stackrel{\mathrm{D}}{\longrightarrow}\left(\begin{bmatrix} {\mathcal X}_{t,1} \\ a_{2,1} \int_0^t {\mathcal X}_{s,1} \, \mathrm{d} s \\[1mm] a_{3,2}a_{2,1} \int_0^t\left(\int_0^r {\mathcal X}_{s,1} \, \mathrm{d} s\right)\,\mathrm{d} r \end{bmatrix}\right)_{t\in\mathbb{R}_+} \quad \text{as $n \to \infty$.} \]

By Itô’s formula, the limit process above is the pathwise unique strong solution of the SDE (3.8) with initial value $[{\mathcal X}_{0,1},{\mathcal X}_{0,2},{\mathcal X}_{0,3}]^{\top}={\boldsymbol{0}}$ , and thus we get the statement of Theorem 3.4.