1. Introduction

As a significant extension of the branching process in a random environment (see [Reference Grama, Liu and Miqueu7, Reference Grama, Liu and Miqueu8, Reference Li, Hu and Liu15, Reference Li, Liu, Gao and Wang18, Reference Wang, Li, Liu and Liu24, Reference Wang, Liu, Li and Liu25] and their references), the branching process with immigration in a random environment (BPIRE) has received extensive attention, Bansaye [Reference Bansaye1] investigated BPIRE by studying a model of cell contamination. Kesten [Reference Kesten, Kozlov and Spitzer13] obtained the limiting distribution of random walks in random environments by using branching processes with one immigration at each generation in an i.i.d. environment. Wang and Liu have obtained the almost surely convergence, L^p convergence, the conditional moments, the quenched moments, the harmonic moments, the exponential decay rate, and the L^p convergence rate under the annealed law about $(W_n);$ and the nondegeneracy, the existence of the p-th moments and the harmonic moments for its limit $W;$ central limit theorem (CLT), the large and moderate deviation principles, and the Berry–Esseen bound for $\log Z_{n}$ [Reference Wang and Liu26–Reference Wang and Liu28]. Wang et al. provide the Cramér’s large deviation expansion for $\log Z_{n}$ [Reference Wang, Liu and Fan29]. Li and Huang [Reference Li and Huang16] investigated a polynomial convergence rate of the submartingale to its limit on BPIRE, and the almost surely convergence rate for a submartingale associated with branching process in a varying environment. In [Reference Li, Huang and Peng17], Li et al. considered the convergence rate in probability or distribution, and two forms of the CLTs about $(W_n).$ Huang et al. [Reference Huang, Li and Xiang12] considered the rate of convergence of the CLT under a moment condition of order $2+\delta$, with fixed $\delta \in(0,1]$. Huang et al. [Reference Huang, Wang and Wang10, Reference Huang, Wang and Wang11] showed the moments and the harmonic moments of Z_n, the large deviation principle and large deviations for $\log Z_{n},$ and described the decay rates of n-step transition probabilities. For the subcritical and critical cases (with multi-type), Key [Reference Key14] demonstrated the convergence to a limit distribution. Roitershtein [Reference Roitershtein21] investigated CLTs and strong laws of large numbers for the partial sums of this process. Additionally, Vatutin [Reference Vatutin22] applied a multi-type BPIRE to study polling systems with random service regimes.

Despite these contributions, there is no result for comparison on the criticality parameters for two supercritical BPIRE, which hinders their practical application. The objective of the paper is to fill this gap.

Let $(\xi_1, \xi_2)^T =( (\xi_{1,n}, \xi_{2,n})^T)_{n\geq 0}$ be a sequence of i.i.d. two-dimensional random vectors, where T is the transport operator and $ ( \xi_{1,n}, \xi_{2,n})^T \in \mathbb{R}^2$ stands for the random environment at generation n. Thus, $( \xi_{1,n}, \xi_{2,n})_{n\geq 0}^T$ are independent random vectors, but notice that for given n, $\xi_{1,n}$ and $\xi_{2,n}$ may not be independent. For any $n\in \mathbb N$ and $i=1,2$, each realization of $\xi_{i,n}$ corresponds to two probability distributions on $\mathbb{N}=\{0,1,2,\cdots\}$: one is the offspring distribution denoted by

\begin{equation*} p(\xi_{i,n})=\{p_{k}(\xi_{i,n}):k \in \mathbb{N} \},\text{where }p_{k}(\xi_{i,n})\geq 0,\quad \sum_{k}p_{k}(\xi_{i,n})=1,\end{equation*}

the other is the distribution of the number of immigrants denoted by

\begin{equation*}\hat{p}(\xi_{i,n})=\{\hat{p}_{k}(\xi_{i,n});k \in\mathbb{N}\},\text{where }\hat{p}(\xi_{i,n})\geq0,\quad \sum_{k}\hat{p}_{k}(\xi_{i,n})=1.\end{equation*}

Let $\{Z_{1,n}, n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two branching processes with immigration in the random environments $\xi_{1,n}$ and $\xi_{2,n}$, respectively. Then, $\{Z_{1,n}, n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ can be described as follows: for $n \geq 0,$

\begin{equation*}Z_{1,0}=1,\ \ \ \ Z_{1,n+1} =Y_{1,n} + \sum_{i=1}^{Z_{1,n}} X_{1, n,i},\ \ \ \ Z_{2,0}=1,\ \ \ \ Z_{2,n+1} =Y_{2,n} + \sum_{i=1}^{Z_{2,n}} X_{2,n,i} , \end{equation*}

where $X_{1,n,i}$ and $X_{2,n,i}$ are the number of offspring of the i-th individual in generation n with environments $\xi_{1,n}$ and $\xi_{2,n}$, respectively. $ Y_{1,n} $ and $ Y_{2,n} $ are the number of new immigrants in the n-th generation with environments $\xi_{1,n}$ and $\xi_{2,n}$. Given $(\xi_{1,n},\, \xi_{2,n})^T$, the random variables $\{X_{1, n,i},X_{2, n,i},i\geq 1\}$ and $\{Y_{1,n},Y_{2,n}\}$ are mutually independent.

Let $(\Gamma, \mathbb P_{\xi})$ be the probability space under which the process is defined when the environment ξ is given. The total probability space can be formulated as the product space $(\Gamma\times \Theta^{\mathbb N}, \mathbb P)$, with $\mathbb P(dx, d\xi) =\mathbb P_{\xi}(dx)\tau(d\xi).$ Usually, the conditional probabilities $\mathbb{P}_{\xi_1}$ and $\mathbb{P}_{\xi_2}$ are called the quenched laws, while the total probability $\mathbb{P}$ is called the annealed law. We further define two laws $\mathbb P_{\xi_i,Y_i}$, $i=1,2$, which denote the conditional probabilities of $\mathbb P$ given $(\xi_i, Y_i)$ where $Y_i=(Y_{i,0},Y_{i,1},...), i=1,2$. Additionally, we denote $\mathbb{P}_{\xi_1 ,\xi_2}$ may be considered to be the conditional probability when the environment $(\xi_1,\, \xi_2)^T$ is given, and τ is the joint law of the environment $(\xi_1,\, \xi_2)^T$. Then,

\begin{equation*}\mathbb{P}(dx_1, dx_2, dy_1, dy_2)=\mathbb{P}_{\xi_1, \xi_2} (dx_1, dx_2)\tau (dy_1, dy_2)\end{equation*}

is the joint law of the two branching processes in random environment. In the sequel, the expectation with respect to $\mathbb P_{\xi_1,\xi_2}$ $(\text{resp. }\mathbb P_{\xi_i,Y_i}, \mathbb P_{\xi},\mathbb P)$ will be denoted by $\mathbb E_{\xi_1,\xi_2}$ $(\text{resp. }\mathbb E_{\xi_i,Y_i}, \mathbb E_{\xi},\mathbb E)$.

We define, for any $ n\ge 0 $ and $ a\ge 0 $,

\begin{equation*} m_{1,n}^{(p)} \left ( a \right ) = \sum_{k=0}^{\infty} k^p \, p_k(\xi_{1,n} ),\ \ \quad m_{2,n}^{(p)}\left ( a \right ) = \sum_{k=0}^{\infty} k^p \, p_k(\xi_{2,n} ),\end{equation*}

\begin{equation*} \quad \Pi_{1,n} = \prod_{i=0}^{n-1} m_{1,i}, \ \ \quad \Pi_{2,n} = \prod_{i=0}^{n-1} m_{2,i},\end{equation*}

with the convention that $ \Pi_{1,0} = \Pi_{2,0} = 1$. Moreover

\begin{equation*}\hat{m}_{1,n}^{(p)}(a)=\sum_{k=0}^{\infty}k^{a}\hat{p}_{k}(\xi_{1,n}), \quad \hat{m}_{2,n}^{(p)}(a)=\sum_{k=0}^{\infty}k^{a}\hat{p}_{k}(\xi_{2,n}).\end{equation*}

Clearly, $(m_{1,n}^{(p)})_{n\geq 0}$ and $(m_{2,n}^{(p)})_{n\geq 0}$ are two sequences of i.i.d. random variables and we denote

\begin{align*} X_{1,n} &= \log m_{1,n} ,\ \ \ \ \ \ X_{2,n} = \log m_{2,n}, \ \ \ \ \ \ \mu_1 = \mathbb{E} \log m_{1,0}, \ \ \ \ \ \ \ \ \ \mu_2 = \mathbb{E} \log m_{2,0}, \\ \sigma_1 ^2 &= \textrm{Var}(\log m_{1,0} ) , \ \ \ \ \ \ \sigma_2 ^2 = \textrm{Var}( \log m_{2,0}),\ \ \ \ \displaystyle \rho = \frac{\textrm{Cov}(X_{1,n}, X_{2,n})}{\sigma_1\sigma_2 } , \end{align*}

where µ ₁ and µ ₂ are known as the criticality parameters for BPIREs $\{Z_{1,n}, n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$, respectively. In particular, if ξ ₁ and ξ ₂ are independent, we have ρ = 0. To avoid the environments ξ ₁ and ξ ₂ are degenerate, we assume that $ 0 \lt \sigma_1, \sigma_2 \lt \infty.$ For $l=1,2,$ to establish some limit theorems on $ Z_{l,n} $ and the fundamental submartingale, we shall use the decomposition of $ Z_{l,n}$, similar to the approach used in [Reference Wang and Liu26].

For simplicity, we will primarily concentrate on the case of $ Z_{1,n}. $ To include the immigrants in the family tree, we introduce one particle at each time n which we call eternal particle, denoted by $0_{0},0_{1},0_{2},\cdots$ with $0_{n}:=0_{n-1}0$ (the juxtaposition of 0₀ with n times 0), and we consider that the $Y_{1,n}$ immigrants are direct children of 0_n. To form a tree, we also consider that each $0_{n+1}$ is a direct child of 0_n. Let $ E=\left \{0_{k} : k \gt 0 \right \} $ represent the set of all virtual particles introduced and assume that the $ Y_{1,n} $ particles moved into the n + 1 generation are the descendants of the virtual particle 0_n introduced in the n-th generation. To construct a complete family tree, we assume that the virtual particle $ 0_{n+1} $ introduced in the n + 1 generation is also the offspring of the virtual particle 0_n introduced in the n-th generation.

To enhance accessibility, we use “∼” to represent the pedigree with the initial particle ϕ without the immigrating particle, and “ $ \wedge $” to represent the pedigree with the initial particle 0₀ including the immigrated particle. Therefore, the $ \tilde{Z} _{1,n} $ represents the branching process in the random environment excluding the immigrating particles, the other $ \hat {Z } _{1,n} $ represents the branching process including the immigrating particles in the random environment, then

\begin{equation*} {Z } _{1,n}=\tilde {Z } _{1,n}^{\left ( \phi \right ) } +\hat{Z} _{1,n}^{\left ( 0_{0} \right ) } -1 ,n\ge 0. \end{equation*}

Set

\begin{equation*} W_{1,n}=\frac{Z_{1,n}}{\Pi_{1,n}}, \ \ \tilde {W}_{1,n}^{\left (\phi\right )}=\frac{\tilde{Z}_{1,n} }{\Pi _{1,n} } \ \ and \ \ \hat{W}_{1,n}^{\left (0_{0}\right )}=\frac{\hat {Z}_{1,n}}{\Pi_{1,n}},\end{equation*}

it is obvious that

(1.1)\begin{eqnarray} {W } _{1,n}=\tilde {W } _{1,n}^{\left (\phi\right )}+\hat{W} _{1,n}^{\left (0_{0}\right )}-\Pi _{1,n}^{-1} . \end{eqnarray}

The sequence $ \tilde {W } _{1,n}^{\left ( \phi \right ) } $ is the well-known martingale associated with the branching process $ \tilde{Z}_{1,n} $ (without immigration) in a random environment, and its asymptotic properties have been extensively studied. We will break down the branching processes with immigration, which begin with an eternal particle $0_{n} \in E$, in terms of branching processes (without immigration) in random environment, we have

\begin{equation*} \hat{W} _{1,n}^{\left ( 0_{0} \right ) }=\frac{1}{m_{1,0} } \hat{W} _{1,n-1}^{\left ( 0_{1} \right ) }+\frac{1}{m_{1,0} } \sum_{i=1}^{Y_{1,0} } \hat{W} _{1,n-1}^{\left ( 0_{0}i \right ) }. \end{equation*}

For the case of a single supercritical BPIRE, denoted by $\{Z_{1,n}, n\geq 0\}$, the normal approximation has been extensively studied. Given the additional conditions $ \mathbb{E}\left ( \frac{Z_{1,0} }{m_{1,0}} \right )^{p} \lt \infty $ and $ \mathbb{E}\left ( \frac{Y_{1,0} }{m_{1,0}} \right ) ^{p} \lt \infty $ for a constant $ p \gt 1,$ and $ \mathbb{E} X_{1,0}^{2+\delta} \lt \infty $ for a constant $\delta \in (0, 1]$. Wang et al. have derived the following Berry–Esseen bound for $\log Z_{1,n}$ in their work [Reference Wang and Liu27]:

(1.2)\begin{equation} \sup_{x \in R} \Big|\mathbb{P}\big( \frac{\log Z_{1,n}-n\mu_{1} }{\sigma \sqrt{n} }\leq x \big) - \Phi(x) \Big| \leq \frac{C} { n^ {\delta/2} }, \end{equation}

where $\Phi(x)$ is the standard normal distribution function.

Assume Cramér’s condition $ \mathbb{E}e^{\lambda _{0} X_{1,0} } \lt \infty $ for a constant $ \lambda_{0} \gt 0,$ and $\mathbb{E}\left (\frac{Z_{1,0}^{p}}{m_{1,0}}\right ) \lt \infty, \mathbb{E}\left (\frac{Y_{1,0}^{p}}{m_{1,0}}\right ) \lt \infty$ for a constant $ p \gt 1,$ Wang et al. [Reference Wang, Liu and Fan29] also have established the following Cramér’s large deviation expansion: for $0 \le x=o\left ( n \right ) ,n\to \infty $,

(1.3)\begin{eqnarray} \Big| \log \frac{\mathbb{P}\big( \frac{\log Z_{1,n}-n\mu_{1}}{\sigma \sqrt{n} } \leq x \big)} {1- \Phi(x) } \Big| \leq C \frac{1+ x^ {3}}{\sqrt{n}} \end{eqnarray}

where C is positive constant. For instance, when the parameter σ ₁ is known, they can be applied to construct confidence intervals for estimating the criticality parameter µ ₁. This estimation is formulated by considering both the observation $Z_{1,n}$ and the generation n, providing a more precise understanding of the process.

Although the limit theorems for a single supercritical BPIRE have been extensively studied, there currently exists no comparative result concerning the criticality parameters for two supercritical BPIREs. The objective of the paper is to fill this gap. We begin by considering the following common hypothesis testing:

\begin{equation*}H_0: \mu_1-\mu_2=0 \ \ or \ \ H_1: \mu_1 -\mu_2 \ne 0.\end{equation*}

When µ ₁ and µ ₂ represent the means of two independent populations, this form of hypothesis testing has been considered by Chang et al. [Reference Chang, Shao and Zhou3]. Within their work, they have established Cramér type moderate deviations. In this paper, we are interested in the case where µ ₁ and µ ₂ are two criticality parameters of BPIREs. By the law of large numbers, $\frac{1}{n} \log Z_{1,n} \to \mu_1$ and $\frac{1}{m} \log Z_{2,m} \to \mu_2$ converge in probability as $m,n\to\infty$, respectively. Therefore, to test the hypothesis, it is essential to estimate the asymptotic distribution of the random variable $\frac{1}{n} \log Z_{1,n} - \frac{1}{m} \log Z_{2,m} $, this estimation is central to the main purpose of this paper. Observe that the expression $\frac{1}{n} \log Z_{1,n} - \frac{1}{m} \log Z_{2,m} $ has an asymptotic distribution equivalent to $\frac{1}{n}\sum_{k=1 }^{n} X_{1,k} - \frac{1}{m}\sum_{k=1 }^{m} X_{2,k}$. When ξ ₁ and ξ ₂ are independent, both $\sum_{k=1 }^{n} X_{1,k} $ and $ \sum_{k=1 }^{m} X_{2,k} $ are sums of i.i.d. random variables.

In this paper, we always assume $ l=1,2, $

(1.4)\begin{equation} \mathbb{E} \log^{+} \frac{Y_{l,0}}{m_{l,0}} \lt \infty \ \ and \ \ \mathbb{E}(\log m_{l,0}) \gt 0 \end{equation}

which means that the process is supercritical. We assume that the following conditions hold:

(1.5)\begin{equation} \mathbb{E}\bigg[ \frac{Z_{1,1}}{m_{1,0}} \log^+ Z_{1,1} + \frac{Z_{2,1}}{m_{2,0}} \log^+ Z_{2,1} \bigg] \lt \infty, \end{equation}

write $ \log ^+ x = \max\{\log x , 0 \}. $ From Grama et al. [Reference Grama, Liu and Miqueu7], it can be inferred that under the conditions (1.4) and (1.5), $ {W} _{n} $ converges almost surely to a non-negative random variable $W.$ Additionally, we assume the following condition:

(1.6)\begin{equation} p_0(\xi_{1,0})=p_0(\xi_{2,0} ) =0, \ \ \ \ \ \ \ a.s. \end{equation}

which ensures that the random walk has positive increments and states that each individual has at least one offspring. Assumptions (1.5) and (1.6) imply that the processes $(Z_{1,n}, n\geq 0)$ and $(Z_{2,m}, m\geq 0)$ are both supercritical and satisfy $\mu_1, \mu_2 \gt 0$ and $Z_{1, n} \to \infty, Z_{2, m} \to \infty $.

Define

\begin{align*} R_{m,n}&:= \frac{\frac{1}{n} \log Z_{1,n} - \mu_1 - \frac{1}{m} \log Z_{2,m} + \mu_2 }{V_{m,n,\rho}},\\ V_{m,n,\rho} &= \sqrt{\frac1n \sigma_1^2 + \frac1m \sigma_2^2 -2 \rho \sigma_1 \sigma_2 \frac{m \wedge n}{m \, n} \ } , \ \ n, m \in \mathbb{N}.\end{align*}

Throughout the paper, we assume either

\begin{equation*} \rho \in [-1, 1) \ \ \ \ \textrm{or} \ \ \ \ \rho =1 \ \text{but} \ \sigma_1 \ne \sigma_2 .\end{equation*}

The final condition guarantees that

\begin{equation*}\frac1n \sigma_1^2 + \frac1m \sigma_2^2 -2 \rho \sigma_1 \sigma_2 \frac{m \wedge n}{m \, n}\end{equation*}

is in order of $\frac{1}{m \wedge n} $ as $m,n \to \infty. $ Clearly, it is easy to see that if $ m \le n ,$

\begin{equation*}\frac1n \sigma_1^2 + \frac1m \sigma_2^2 -2 \rho \sigma_1 \sigma_2 \frac{m \wedge n}{m \, n} \ = (\frac1m -\frac1n)\sigma _{2}^{2} +\frac{\sigma _{1}^{2}-2\rho\sigma _{1}\sigma _{2}+\sigma _{2}^{2} }{n}\asymp \frac{1}{m}. \end{equation*}

We now introduce our main results. First, Theorem 2.1 presents the CLT for $R_{m,n}:$ for all $x \in \mathbb{R},$ it holds

(1.7)\begin{equation} \lim_{m \wedge n \rightarrow \infty }\mathbb{P}\big( R_{m,n} \leq x \big) = \Phi(x). \end{equation}

Second, under some moment conditions, Theorem 2.2 gives a non-uniform Berry–Esseen bound for $R_{m,n}$: for any $\delta' \in (0, \delta)$ and all $x \in \mathbb{R},$

(1.8)\begin{equation} \bigg|\mathbb{P}\big( R_{m,n} \leq x \big) - \Phi(x) \bigg| \leq \frac{ C }{ (m\wedge n)^{\delta/2} }\frac{ 1 }{ 1+|x|^{1+\delta'} }. \end{equation}

According to Lemma 4.3 and (3.16) in the paper, under the given conditions, we conclude that $R_{m,n}$ only has a finite moment of order $1+\delta'$. This explains why the non-uniform Berry–Esseen bound exhibits an order of $\displaystyle |x|^{-1-\delta'}$ as $x \rightarrow \infty$, instead of an order of $\displaystyle |x|^{-2-\delta}$. In particular, we have $\frac{1}{m} \log Z_{2,m} \rightarrow \mu_2$ in probability when $m \rightarrow \infty,$ which leads to $R_{m,n} \rightarrow \frac{\log Z_{1,n} - n \mu_1 \ }{ \sigma_1 \sqrt{n}}$ in probability. Thus, inequality (1.8) implies that

\begin{equation*} \sup_{x\in \mathbb{R}} \bigg|\mathbb{P}\Big( \frac{\log Z_{1,n} - n \mu_1 \ }{ \sigma_1 \sqrt{n}} \leq x \Big) - \Phi(x) \bigg| \leq \frac{ C }{ (m\wedge n)^{\delta/2} }\frac{ 1 }{ 1+|x|^{1+\delta'} }, \nonumber \end{equation*}

which improves the Berry–Esseen bound (1.2) by adding a factor $\frac{ 1 }{ 1+|x|^{1+\delta'} } .$

Third, we establish Cramér’s moderate deviations. Assuming conditions A3, A4, and A5 are satisfied, Theorem 2.3 demonstrates that for all $0 \leq x \leq c^{-1} \sqrt{m \wedge n } $,

(1.9)\begin{equation} \Bigg| \log \frac{\mathbb{P}\big( R_{m,n} \geq x \big)}{1-\Phi(x)} \Bigg| \leq C \frac{1+x^3 }{ \sqrt{m \wedge n} \ }. \end{equation}

When $m\rightarrow \infty,$ it is easy to see that (1.9) holds with $R_{m,n}$ replaced by $\frac{\log Z_{1,n} - n \mu_1 \ }{ \sigma_1 \sqrt{n}} $. Therefore, our results reconfirm the Cramér’s moderate deviations (1.3) as initially established by Wang et al. To conclude, we explore the creation of confidence intervals for $\mu_1 - \mu_2$ as an application of our finding.

We now explain briefly the organization of this paper. In Section 2, we present our main results. Some applications are demonstrated in Section 3. The proofs of some results in Section 2 are given in Section 3.2.

Additionally, the symbols c and C are used to represent a small positive constant and a large positive constant, respectively. Their values may vary from line to line. For two sequences of positive numbers $(a_{n} )_{n\ge 1}$ and $(b_{n} )_{n\ge 1}$, we write $a_n \asymp b_n$ if there exists a positive constant C such that for all n, it holds $C^{-1}b_n \leq a_n \leq C b_n$.