2.1. Compound Poisson cluster point processes

Let $\{X_n\}_{n\geq 1}$ be the points of a Poisson process on $\mathbb R^d$ , $d\geq 1$ , with a locally integrable intensity function $\lambda\;:\;\mathbb{R}^d\to [0,\infty)$ , and let $\{Z_n(\cdot,\cdot)\}_{n\geq 1}$ be a sequence of independent and identically distributed simple point processes on ${\mathbb R}^d \times {\mathbb R}$ , independent of $\{X_n\}_{n\geq 1}$ . More concretely, for $(C_1,C_2) \in \mathcal B ({\mathbb R}^d) \times \mathcal B ({\mathbb R})$ , $Z_n(C_1,C_2)$ counts the number of points of the nth point process that fall in $C_1$ and whose marks are in $C_2$ . For each $n\geq 1$ , we denote the points of $Z_n(\cdot, \cdot)$ by $\{(Y_{n,k},M_{n,k})\}_{k\geq 0}$ , and we assume that $Y_{n,0}\;:\!=\; \textbf {0}$ (which implies $Z_n(\{\textbf{0}\},{\mathbb R})\;:\!=\;1$ ) and that the sequence $\{M_{n,k}\}_{k\geq 0}$ is independent of $\{Y_{n,k}\}_{k\geq 0}$ . Furthermore, we suppose that the random variables $\{M_{n,k}\}_{n\geq 1,\,k\geq 0}$ are independent and identically distributed. Throughout the paper we denote by M the generic random variable $M_{n,k}$ .

One naturally interprets the first component of each point of $Z_n(\cdot,\cdot)$ as a ‘location’, and the second component as a ‘mark’ which describes some characteristic of the location to which it is attached. Hereafter, for $n\geq 1$ , we consider the point processes $\theta_{X_n}Z_n(\cdot,\cdot)\equiv\{(X_n+Y_{n,k},M_{n,k})\}_{k\geq 0}$ .

For arbitrarily fixed $n\geq 1$ and $C \in \mathcal B ({\mathbb R}^d)$ , we define the random variable

(2) \begin{equation}\upsilon(Z_n)(C)\;:\!=\;\sum _{k=0}^{Z_n(C,{\mathbb R})-1}M_{n,k},\end{equation}

which aggregates the marks attached to the locations that fall in C. It turns out that the random variable, say V(C), which aggregates all the marks attached to the points which fall in C of the Poisson cluster point process

\[N\equiv\bigcup_{n\geq 1}\{X_n+Y_{n,k}\}_{k\geq 0}\]

is a Poisson shot noise random variable. Indeed,

(3) \begin{equation}V(C)\;:\!=\;\sum_{n\geq 1}\sum _{k=0}^{\theta_{X_n}Z_n(C,{\mathbb R})-1}M_{n,k}=\sum_{n\geq 1}\upsilon(\theta_{X_n}Z_n)(C)=\sum_{n\geq 1}\upsilon(Z_n)(C-X_n)\end{equation}

is a random variable of the form (1) with $H(C-x,z)\;:\!=\;v(z)(C-x)$ , $x\in\mathbb{R}^d$ , $z\in\textbf{Z}\;:\!=\;\textbf{N}_{\mathbb{R}^d\times\mathbb{R}}$ . Here $\textbf{N}_{\mathbb R^d\times\mathbb R}$ denotes the space of $\sigma$ -finite counting measures on $(\mathbb R^d\times\mathbb{R},\mathcal B(\mathbb R^d)\otimes\mathcal B(\mathbb R))$ equipped with the usual $\sigma$ -field (see Section 4 for details), and

(4) \begin{equation}v(z)(C)\;:\!=\;\sum_{k\geq 0}\textbf{1}_C(y_k)m_k,\quad\text{for $z\equiv\{(y_k,m_k)\}_{k\geq 0}$.}\end{equation}

Note that if $M_{n,k}\;:\!=\;1$ for every $n\geq 1$ and $k\geq 0$ , then the random variable

(5) \begin{equation}N(C)\;:\!=\;V(C)= \sum_{n\geq 1}Z_n(C-X_n,{\mathbb R})\end{equation}

equals the number of points of the Poisson cluster point process N which fall in $C\in\mathcal{B}({\mathbb R}^d)$ .

2.2. Generalized Hawkes processes and generalized compound Hawkes processes

Let $N\equiv\{N(C)\}_{C\in\mathcal{B}({\mathbb R}^d)}$ be the Poisson cluster point process defined by (5). We shall refer to N as a generalized Hawkes process if the random variable $Z\;:\!=\;Z_1({\mathbb R}^d,{\mathbb R})$ is distributed as the total progeny of a sub-critical Galton–Watson process with one ancestor. It is worthwhile to note the following:

1. (i) Classical Hawkes processes on $(0,\infty)$ (respectively, on $\mathbb R$ ) with parameters $(\lambda,g)$ , introduced in the seminal papers [Reference Hawkes21, Reference Hawkes and Oakes22], are particular examples of generalized Hawkes processes. Indeed, they are Poisson cluster point processes defined as follows. (1) The process of cluster centers $\{X_n\}_{n\geq 1}$ is a Poisson process on $(0,\infty)$ (respectively, on $\mathbb R$ ) with constant intensity equal to $\lambda>0$ . (2) The points of the cluster $\theta_{X_n}Z_n(\cdot,{\mathbb R})$ are partitioned into generations and generated recursively as follows. The ancestor constitutes the generation 0 of the cluster and is located at $X_n$ . Given $X_n$ , the ancestor generates points of the first generation of the cluster according to a non-homogeneous Poisson process on $(X_n,\infty)$ with intensity function $g(\cdot-X_n)$ , where $g\;:\;\mathbb R\to [0,\infty)$ is a measurable function which is null on $(\!-\!\infty,0]$ and such that $h\;:\!=\;\int_0^\infty g(x)\mathrm{d}x<1$ . In turn, given the points of the first generation of the cluster, a point of this generation, which is located at X, generates points of the second generation of the cluster according to a non-homogeneous Poisson process on $(X,\infty)$ with intensity function $g(\cdot-X)$ ; and so on and so forth. Note that $Z_n(\mathbb R,{\mathbb R})=\theta_{X_n}Z_n(\mathbb R,{\mathbb R})=\theta_{X_n}Z_n([X_n,\infty),{\mathbb R})=Z_n([0,\infty),{\mathbb R})$ is distributed as the total progeny of a sub-critical Galton–Watson process with one ancestor and Poisson offspring law with mean h.

2. (ii) Spatial Hawkes processes on ${\mathbb R}^d$ , $d\geq 1$ , with parameters $(\lambda,g)$ , introduced in [Reference Brémaud, Massoulié and Ridolfi9] and further studied in [Reference Møller and Torrisi34], are also particular examples of generalized Hawkes processes. Indeed, they are Poisson cluster point processes defined as follows. (1) The process of cluster centers $\{X_n\}_{n\geq 1}$ is a Poisson process on $\mathbb R^d$ , $d\geq 1$ , with constant intensity equal to $\lambda>0$ . (2) The points of the cluster $\theta_{X_n}Z_n(\cdot,{\mathbb R})$ are partitioned into generations and generated recursively as follows. The ancestor constitutes the generation 0 of the cluster and is located at $X_n$ . Given $X_n$ , the ancestor generates points of the first generation of the cluster according to a non-homogeneous Poisson process on $\mathbb R^d$ with intensity function $g(\cdot-X_n)$ , where $g\;:\;\mathbb R^d\to [0,\infty)$ is a measurable function such that $h\;:\!=\;\int_{\mathbb R^d}g(x)\mathrm{d}x<1$ . In turn, given the points of the first generation of the cluster, a point of this generation, which is located at X, generates points of the second generation of the cluster according to a non-homogeneous Poisson process on $\mathbb R^d$ with intensity function $g(\cdot-X)$ ; and so on and so forth. Note that $\theta_{X_n}Z_n(\mathbb R^d,{\mathbb R})=Z_n(\mathbb R^d,{\mathbb R})$ is distributed as the total progeny of a sub-critical Galton–Watson process with one ancestor and Poisson offspring law with mean h.

Note that, according to these definitions, classical Hawkes processes on $\mathbb R$ are different from spatial Hawkes processes on $\mathbb R$ .

The collection of random variables $V\equiv\{V(C)\}_{C\in\mathcal{B}({\mathbb R}^d)}$ , where V(C) is defined by (3), will be called a generalized compound Hawkes process if the random variable Z is distributed as the total progeny of a sub-critical Galton–Watson process with one ancestor. Note that V(C) aggregates the marks attached to the points of a generalized Hawkes process which fall in C.

2.3. Interference in wireless communication

Consider the following simple model of wireless communication, which accounts for interference effects that arise when several nodes transmit at the same time. Suppose that transmitting nodes (e.g., antennas) are located according to $\{X_n\}_{n\geq 1}$ , a Poisson process on the plane with intensity function $\lambda(\cdot)$ —i.e., $X_n$ is the location of node n—and denote by $Z_n\in (0,\infty)$ the signal power of the transmitting node n. Suppose that the sequence $\{Z_n\}_{n\geq 1}$ is independent of the Poisson process, and that the random variables $Z_n$ , $n\geq 1$ , are independent and identically distributed. Assume that a receiver is located at the origin $\textbf{0}\in\mathbb R^2$ and that a new transmitter is added at $x\in\mathbb R^2$ and has signal power $y\in (0,\infty)$ . Suppose that the physical propagation of the signal is described by a measurable positive function $A\;:\;\mathbb R^2\to (0,\infty)$ , which gives the attenuation or path loss of the signal power. For simplicity, we assume that random fading (due to occluding objects, reflections, multipath interference, etc.) is encoded in the random variables $Z_n\in (0,\infty)$ . Thus, $Z_n A(X_n)$ is the power received at the origin from the transmitting node at $X_n$ , and the total interference at the origin, due to simultaneous transmissions, is equal to

\[I(\{\textbf{0}\})\;:\!=\;\sum_{n\geq 1}Z_n A(X_n).\]

Note that this is a Poisson shot noise random variable of the form (1). Indeed, let $H\;:\;\mathcal{B}({\mathbb R}^2)\times (0,\infty)\to (0,\infty)$ be a mapping which, restricted to ${\mathbb R}^2\times (0,\infty)$ , coincides with $\widetilde{H}(x,z)\;:\!=\;zA(\!-x)$ . Then

\[S(\{\textbf{0}\})=\sum_{n\geq 1}H(\{\textbf{0}\}-X_n,Z_n)=\sum_{n\geq 1}\widetilde{H}(\!-X_n,Z_n)=I(\{\textbf{0}\}).\]

We refer the reader to [Reference Baccelli and Błaszczyszyn2, Reference Baccelli and Błaszczyszyn3] for more insight into this model, and limit ourselves to observing that the receiver at the origin can decode the signal of power $y\in (0,\infty)$ from the transmitter at $x\in\mathbb R^2$ if and only if the signal-to-interference-plus-noise ratio (SINR) is bigger than a given threshold, i.e.,

\[\mathrm{SINR}\;:\!=\;\frac{yA(x)}{I(\{\textbf{0}\})+w}\geq\tau,\]

where w is e.g. a thermal noise near the receiver at the origin and $\tau$ is the given threshold.

4.1. Bounds on the Wasserstein and Kolmogorov distances between the law of a first chaos on the Poisson space and the standard normal distribution

Let X and Y be two real-valued random variables defined on $(\Omega,\mathcal{F},\mathbb P)$ . The Wasserstein and Kolmogorov distances between the law of X and the law of Y, written $d_W(X,Y)$ and $d_K(X,Y)$ , respectively, are defined by

\[d_W(X,Y)\;:\!=\;\sup_{g\in\mathrm{Lip}(1)}|\mathbb E[g(X)-g(Y)]|\]

and

\[d_K(X,Y)\;:\!=\;\sup_{x\in{\mathbb R}}|\mathbb{P}(X\leq x)-\mathbb{P}(Y\leq x)|.\]

Here, $\mathrm{Lip}(1)$ denotes the set of Lipschitz functions $g\;:\;{\mathbb R}\to{\mathbb R}$ with Lipschitz constant less than or equal to 1. We recall that throughout this paper, G denotes a random variable distributed according to the standard normal law.

Theorem 1. Let $f\in L^2(\alpha)$ be such that $\|f\|_{L^2(\alpha)}=1$ . Then

\[d_W(I(f),G)\leq\int_{A}|f(a)|^3\alpha(\mathrm{d}a)\]

and

\begin{align*}d_K(I(f),G)\leq & \left(1+\frac{1}{2}\max\left\{4,\left[4\int_A|f(a)|^4\alpha(\mathrm{d}a)+2\right]^{1/4}\right\}\right)\int_A|f(a)|^3\alpha(\mathrm{d}a)\\[5pt] &+\left(\int_A|f(a)|^4\alpha(\mathrm{d}a)\right)^{1/2}.\nonumber\end{align*}

Proof. We refer the reader to [Reference Last and Penrose29] for all the notions of stochastic analysis on the Poisson space used in this proof. Let F be a functional of $\Pi$ , i.e., $F=\mathfrak{f}(\Pi)$ , where $\mathfrak f$ is a real-valued measurable function defined on ${\textbf{N}}_A$ . We recall that the difference operator D is defined by

\[D_a F=\mathfrak{f}(\Pi+\delta_a)-\mathfrak{f}(\Pi),\quad a\in A,\]

where $\delta_a$ is the Dirac measure at $a\in A$ , and that the second difference operator $D^2$ is defined by

\[D_{a_1,a_2}^{2}F=D_{a_1}(D_{a_2}F),\quad a_1,a_2\in A.\]

We also recall that the domain of D, denoted by $\mathrm{dom}(D)$ , is the family of square-integrable random variables $F=\mathfrak{f}(\Pi)$ such that

\[\int_A\mathbb E |D_a F|^2\alpha(\mathrm{d}a)<\infty.\]

Setting $F\;:\!=\;I(f)$ , we have that ${\mathbb E} F = 0$ with $\mathbb{V}\mathrm{ar}(F)=1$ (as follows by applying the isometry formula for Poisson chaoses) and that $F\in\mathrm{dom}(D)$ . Using Theorem 1.1 in [Reference Last, Peccati and Schulte30] we have that

$$d_W(F,G) \leq \gamma_1 + \gamma_2 + \gamma_3,$$

where

\begin{equation*} \begin{cases} \gamma_1 &= 2 \left( \int_{A^3} \left({\mathbb E} [(D_{a_1}F)^2 (D_{a_2}F)^2] \right)^{1/2} \left( {\mathbb E} [(D^2_{a_1,a_3}F)^2 (D^2_{a_2,a_3}F)^2]\right)^{1/2} \alpha^{3}(\mathrm {d}(a_1,a_2,a_3))\right)^{1/2}, \\[5pt] \gamma_2 &=\left(\int_{A^3} {\mathbb E} [(D^2_{a_1,a_3}F)^2 (D^2_{a_2,a_3}F)^2]\alpha^{3}(\mathrm {d}(a_1,a_2,a_3)) \right)^{1/2},\\[5pt] \gamma_3 &= \int_A {\mathbb E}|D_{a}F|^3 \alpha(\mathrm d a). \end{cases}\end{equation*}

Since for $F=I(f)$ we have that $D_a F=f(a)$ and $D_{a_1,a_2}F=0$ , $a_1,a_2\in A$ , the terms $\gamma_1$ and $\gamma_2$ vanish and $\gamma_3= \int_A |f(a)|^3 \alpha( \mathrm d a)$ . Therefore, we obtain the bound on the Wasserstein distance between I(f) and G.

Similarly, Theorem 1.2 in [Reference Last, Peccati and Schulte30] gives

$$d_K(F,G) \leq \left( 1 + \frac{1}{2} \left({\mathbb E} F^4\right)^{1/4} \right) \int_A |f(a)| ^{3} \alpha(\mathrm d a) + \left(\int_A f(a)^4 \alpha(\mathrm d a) \right)^{1/2}.$$

Using Lemma 4.2 in [Reference Last, Peccati and Schulte30] we have

\begin{align*} {\mathbb E} F^4 & \leq \max \left \{256 \left(\int_A f(a)^2 \alpha(\mathrm d a)\right)^{2}, 4 \int_A f(a)^4 \alpha(\mathrm d a )+2 \right \}\\[5pt] &= \max \left \{4^4, 4 \int_A f(a)^4 \alpha(\mathrm d a )+2 \right \} \quad \text{(because $\|f\|_{L^2(\alpha)}=1$),} \end{align*}

which yields the upper bound on the Kolmogorov distance.

4.2. Moderate deviations, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction term for first chaoses on the Poisson space

We start with a definition.

As a preliminary result, we provide moderate deviations, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction term for sequences of first chaoses on the Poisson space.

Theorem 2. Assume the following: (i) $f_\ell\in L^m(\alpha_\ell)$ for any $m\geq 2$ and any $\ell\in\mathbb N$ with $\|f_\ell\|_{L^2(\alpha_\ell)}=1$ for any $\ell\in\mathbb N$ ; (ii) there exist a constant $\gamma\geq 0$ and a positive numerical sequence $\{\Delta_\ell\}_{\ell\in\mathbb N}$ such that

\[\Big|\int_A f_\ell(a)^m\alpha_\ell(\mathrm{d}a)\Big|\leq\frac{(m!)^{1+\gamma}}{\Delta_\ell^{m-2}},\quad\text{for all $m\geq 3$ and $\ell\in\mathbb N$.}\]

Then the sequence $\{I^{(\ell)}(f_\ell)\}_{\ell\in\mathbb N}$ satisfies an $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , a $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , and an $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ .

Proof. We recall that for real-valued random variables $X_1,\ldots,X_m$ , $m\in\mathbb N$ , the joint cumulant of $X_1,\ldots,X_m$ is defined as

\[\mathrm{cum}(X_1,\ldots,X_m)\;:\!=\;(\!-\textbf{i})^m\frac{\partial^m}{\partial t_1\ldots\partial t_m}\log\varphi_{X_1,\ldots,X_m}(t_1,\ldots, t_m)\Big|_{t_1=\ldots=t_m=0},\]

where ${\textbf{i}}$ is the imaginary unit and $\varphi_{X_1,\ldots,X_m}$ is the joint characteristic function of $(X_1,\ldots,X_m)$ . For a real-valued random variable X and $m\in\mathbb N$ we shall write $\mathrm{cum}_m(X)\;:\!=\;\mathrm{cum}(X,\ldots,X)$ for the mth cumulant of X.

For an arbitrarily fixed $\ell\in\mathbb N$ , set $X_\ell\;:\!=\;I^{(\ell)}(f_\ell)$ . Clearly, $\mathbb E X_\ell=0$ and $\mathbb E X_\ell^2=1$ (which is a consequence of the isometry formula for Poisson chaoses). Then the claim follows by the theory developed in [Reference Saulis and Statulevicius40] (see e.g. Proposition 2.1 in [Reference Schulte and Thäle41]; see also [Reference Döring and Eichelsbacher13, Reference Döring, Jansen and Schubert14]) if we prove that

(13) \begin{equation}\mathbb E |X_\ell|^m<\infty \quad\text{for any $\ell\in\mathbb N$ and $m\geq 3$}\end{equation}

and

(14) \begin{equation}\mathrm{cum}_m(X_\ell)=\int_A f_\ell(a)^m\alpha_\ell(\mathrm{d}a)\quad\text{for any $\ell\in\mathbb N$ and $m\geq 3$.}\end{equation}

To this end we are going to apply Theorem 3.6 of [Reference Schulte and Thäle41]. A partition $\sigma$ of $\{1,\ldots,m\}$ , $m\geq 3$ , is a collection $\{B_1,\ldots,B_k\}$ of $1\leq k\leq m$ pairwise disjoint non-empty sets, called blocks, such that $B_1\cup\ldots\cup B_k=\{1,\ldots,m\}$ . The number k of blocks of a partition $\sigma$ is denoted by $|\sigma|$ . Let $J_j\;:\!=\;\{j\}$ , $j\in\{1,\ldots,m\}$ . Letting $\Pi(\textbf{1}_m)$ , $\textbf{1}_m\;:\!=\;(1,\ldots,1)\in{\mathbb R}^m$ , denote the set of all partitions $\sigma$ of $\{1,\ldots,m\}$ whose blocks B are such that $\mathrm{Card}(B\cap J_{j})\leq 1$ for every $j\in\{1,\ldots,m\}$ , we clearly have that $\Pi(\textbf{1}_m)$ is the set of all partitions of $\{1,\ldots,m\}$ . Letting $\widetilde{\Pi}(\textbf{1}_m)$ denote the set of all partitions $\sigma\in\Pi(\textbf{1}_m)$ with $|\sigma|=1$ , we clearly have $\widetilde\Pi(\textbf{1}_m)=\{\{1,\ldots,m\}\}$ , $m\geq 3$ . Letting $\widetilde{\Pi}_{\geq 2}(\textbf{1}_m)$ denote the set of all partitions $\sigma\in\widetilde\Pi(\textbf{1}_m)$ whose blocks have cardinality bigger than or equal to 2, since $m\geq 3$ , we clearly have $\widetilde\Pi_{\geq 2}(\textbf{1}_m)=\widetilde\Pi(\textbf{1}_m)=\{\{1,\ldots,m\}\}$ . We denote by $\Pi_{\geq 2}(\textbf{1}_m)$ , $m\geq 2$ , the family of all partitions $\sigma\in\Pi(\textbf{1}_m)$ , i.e., the family of all partitions $\sigma$ of $\{1,\ldots,m\}$ whose blocks have cardinality bigger than or equal to 2.

For a function $g\;:\;A\to\mathbb R$ , set

\[(\!\otimes_{j=1}^{m}g)(x_1,\ldots,x_m)\;:\!=\;g(x_1)\ldots g(x_m).\]

For $\sigma\in\Pi(\textbf{1}_m)$ , define the function $(\!\otimes_{j=1}^{m}g)_\sigma\;:\;A^{|\sigma|}\to{\mathbb R}$ by replacing in $(\!\otimes_{j=1}^{m}g)(x_1,\ldots,x_m)$ all the variables whose indexes belong to the same block of $\sigma$ by a new common variable. Note that for $\sigma\in\Pi(\textbf{1}_m)$ , $(\!\otimes_{j=1}^{m}g)_\sigma\;:\;A^{|\sigma|}\to{\mathbb R}$ can be represented as

\[(\!\otimes_{j=1}^{m}g)_\sigma(a_1,\ldots,a_{|\sigma|})=\prod_{i=1}^{|\sigma|}g(a_i)^{|B_i|},\quad a_1,\ldots,a_{|\sigma|}\in A,\]

where $B_1,\ldots,B_{|\sigma|}$ are the blocks of $\sigma$ and $|B_i|$ , $i=1,\ldots,|\sigma|$ , is the cardinality of the block $B_i$ . In particular, for $\sigma\in\widetilde\Pi_{\geq 2}(\textbf{1}_m)$ , $(\!\otimes_{j=1}^{m}g)_\sigma(a)\;:\!=\;g(a)^m,$ $a\in A$ , and, for $\sigma\in\Pi _{\geq 2}(\textbf{1}_m)$ ,

\[(\!\otimes_{j=1}^{m}g)_\sigma(a_1,\ldots,a_{|\sigma|})\;:\!=\;\prod_{i=1}^{|\sigma|}g(a_i)^{|B_i|},\quad a_1,\ldots,a_{|\sigma|}\in A,\]

where $B_1,\ldots,B_{|\sigma|}$ are the blocks of $\sigma$ and $|B_i|\geq 2$ for any $i=1,\ldots,|\sigma|$ . Therefore the hypothesis (i) implies the assumptions $(3.4)$ and $(3.5)$ of Theorem 3.6 in [Reference Schulte and Thäle41], and so (13) and (14) hold.

5.1. Gaussian approximation

In this section, we apply Theorem 1 to the standardized random variables T(C), $C\in\mathcal{B}(\mathbb R^d)$ , defined by (7). Note that, for $C\in\mathcal{B}({\mathbb R}^d)$ ,

\[\mathbb E S(C)=\int_{\mathbb R^d} \lambda(x) \mathbb E H(C-x,Z_1) \mathrm d x,\]

and that, by the isometry formula for Poisson chaoses, if $\int_{\mathbb R^d}\lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x<\infty$ , then

\[\mathbb{V}\mathrm{ar}(S(C))=\int_{\mathbb R^d}\lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x.\]

Therefore the random variable T(C) is well defined and finite for any $C \in \mathcal{B}(\mathbb R^d)$ such that

(15) \begin{equation}0 <\int_{\mathbb R^d}\lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x<\infty.\end{equation}

The following theorem holds.

Theorem 3. Let $C \in \mathcal{B}(\mathbb R^d)$ be such that (15) holds.

Then

\[d_W(T(C),G)\leq\frac{\int_{{\mathbb R}^d}\lambda(x)\mathbb{E}|H(C-x,Z_1)|^3\mathrm{d}x}{\left(\int_{{\mathbb R}^d}\lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x\right)^{3/2}}\]

and

\begin{align}&d_K(T(C),G)\nonumber\\[5pt] &\leq\left[1+\frac{1}{2}\max\Biggl\{4,\left[4\frac{\int_{{\mathbb R}^d}\mathbb \lambda(x)\mathbb{E}H(C-x,Z_1)^4\mathrm{d}x}{\left(\int_{{\mathbb R}^d} \lambda(x) \mathbb{E}H(C-x,Z_1)^2\mathrm{d}x\right)^2}+2\right]^{1/4}\Bigg\}\right]\frac{\int_{{\mathbb R}^d}\lambda(x)\mathbb{E}|H(C-x,Z_1)|^3\mathrm{d}x}{\left(\int_{{\mathbb R}^d}\lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x\right)^{3/2}}\nonumber\\[5pt] &\qquad\qquad\qquad+\left(\frac{\int_{{\mathbb R}^d}\mathbb \lambda(x)\mathbb{E}H(C-x,Z_1)^4\mathrm{d}x}{\left(\int_{{\mathbb R}^d} \lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x\right)^2}\right)^{1/2}.\nonumber\end{align}

Proof. By (1) we have

\begin{equation*}T(C)=\frac{1}{\sqrt{\int_{\mathbb R^d}\lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x}}\int_{\mathbb R^d\times\textbf{M}}H(C-x,z)(\mathcal{P}(\mathrm{d}x,\mathrm{d}z)-\lambda(x)\mathrm{d}x\mathbb{Q}(\mathrm{d}z)),\end{equation*}

$C\in\mathcal{B}({\mathbb R}^d)$ . Therefore T(C) belongs to the first chaos of $\mathcal P$ with kernel

\[t(x,z)\;:\!=\;\frac{H(C-x,z)}{\sqrt{\int_{\mathbb R^d}\lambda(x)\mathbb{E}H(C-x,Z_1)^2\mathrm{d}x}}.\]

The claim follows from applying Theorem 1.

5.2. Moderate deviations, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction term

In this section, we apply Theorem 2 to the sequence of standardized random variables $\{T_\ell(C_\ell)\}_{\ell\geq 1}$ , $\{C_\ell\}_{\ell\geq 1}\subset\mathcal{B}(\mathbb R^d)$ , defined by (6).

The following theorem holds.

Theorem 4. Let $\{C_\ell\}_{\ell \in \mathbb N}\subset\mathcal{B}(\mathbb R^d)$ be a sequence of Borel sets such that

(16) \begin{equation}0<\int_{{\mathbb R}^d}\lambda_\ell(x)\mathbb{E}H(C_\ell-x,Z_1^{(\ell)})^2\mathrm{d}x<\infty,\quad \ell\geq 1,\end{equation}

and assume that there exist a non-negative constant $\gamma\geq 0$ and a positive numerical sequence $\{\Delta_\ell\}_{\ell \in \mathbb N}$ such that

(17) \begin{equation}\frac{\int_{{\mathbb R}^d} \lambda_\ell (x) {\mathbb E} |H(C_\ell-x,Z_1^{(\ell)})|^m \mathrm d x}{\left( \int_{{\mathbb R}^d} \lambda_\ell (x) {\mathbb E} H(C_\ell-x,Z_1^{(\ell)})^2 \mathrm d x\right)^{\frac{m}{2}}} \leq \frac{(m!)^{1+\gamma}}{\Delta_\ell ^{m-2}},\quad \text{for all } m\geq 3 \text{ and }\ell\in\mathbb N.\end{equation}

Then the sequence $\{T_\ell(C_\ell)\}_{\ell\geq 1}$ satisfies an $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , a $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , and an $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ .

Proof. Similarly to the proof of Theorem 3, by (6) we have

\begin{equation*}T_\ell(C_\ell)=\frac{1}{\sqrt{\int_{\mathbb R^d}\lambda_\ell(x)\mathbb{E}H(C_\ell-x,Z_1^{(\ell)})^2\mathrm{d}x}}\int_{\mathbb R^d\times\textbf{Z}}H(C_\ell-x,z)(\mathcal{P}_\ell(\mathrm{d}x,\mathrm{d}z)-\lambda_\ell(x)\mathrm{d}x\mathbb{Q}_\ell(\mathrm{d}z)),\end{equation*}

$C_\ell\in\mathcal{B}({\mathbb R}^d)$ . Therefore $T_\ell(C_\ell)$ belongs to the first chaos of $\mathcal P_\ell$ with kernel

\[t_\ell(x,z)\;:\!=\;\frac{H(C_\ell-x,z)}{\sqrt{\int_{\mathbb R^d}\lambda_\ell(x)\mathbb{E}H(C_\ell-x,Z_1^{(\ell)})^2\mathrm{d}x}}.\]

The claim follows by Theorem 2.

6.1. Gaussian approximation

In this section we apply Theorem 3 to the class of standardized compound Poisson cluster point processes $\{W(C)\}_{C\in\mathcal{B}({\mathbb R}^d)}$ with $\{X_n\}_{n\geq 1}$ having a piecewise constant intensity function. In such a case we have more explicit upper bounds on the Wasserstein and Kolmogorov distances, which pave the way to explicit bounds for some classes of generalized compound Hawkes processes (see Corollaries 3 and 5). We recall that Z denotes the random total number of points of the progeny process, i.e. $Z=Z_1({\mathbb R}^d,{\mathbb R})$ , and that M is a generic random variable with the same distribution as the independent and identically distributed marks.

Hereafter, we denote by $\mathrm{Leb}(\cdot)$ the Lebesgue measure on $\mathbb R^d$ .

Corollary 1. Let $(B,C)\in\mathcal{B}(\mathbb R^d)^2$ be such that $0<\mathrm{Leb}(B\cap C)<+\infty$ . If $\lambda(x)=\lambda \boldsymbol 1 _{B}(x)$ for any $x\in{\mathbb R}^d$ and some positive constant $\lambda>0$ , $\mathbb E Z^2<\infty$ , and ${\mathbb E} M^2\in (0,\infty)$ , then

(18) \begin{equation}d_W(W(C),G)\leq \frac{{\mathbb E} |M|^3\mathbb E Z^3}{({\mathbb E} M^2)^{3/2}\sqrt {\lambda \mathrm{Leb}(B\cap C)}}\end{equation}

and

(19) \begin{align}d_K(W(C),G)&\leq\left[1+\frac{1}{2}\max\Biggl\{4,\left[4\frac{{\mathbb E} M^4\mathbb{E} Z^4}{\lambda\mathrm{Leb}(B\cap C)({\mathbb E} M^2)^2}+2\right]^{1/4}\Bigg\}\right]\frac{{\mathbb E} |M|^3\mathbb E Z^3}{({\mathbb E} M^2)^{3/2}\sqrt {\lambda \mathrm{Leb}(B\cap C)}}\nonumber\\[5pt] &\qquad\qquad\qquad\qquad+\left(\frac{{\mathbb E} M^4\mathbb{E} Z^4}{\lambda \mathrm{Leb}(B\cap C)({\mathbb E} M^2)^2}\right)^{1/2}.\end{align}

We point out that many articles in the literature (e.g. [Reference Bacry, Delattre, Hoffmann and Muzy4, Reference Hillairet, Huang, Khabou and Réveillac23, Reference Khabou, Privault and Réveillac26]) consider Hawkes processes with an empty history, that is, with no points in $(\!-\!\infty,0]$ , which corresponds to the piecewise constant intensity function $\lambda(x)=\boldsymbol 1_{[0,+\infty)}(x)$ in Corollary 1.

The proof of Corollary 1 exploits the following lemma, which is proved in Appendix A.

Lemma 1. For any $(B,C)\in\mathcal{B}(\mathbb R^d)^2$ and $m\in\mathbb N$ , we have

(20) \begin{equation}\int_{B} {\mathbb E} |\upsilon(Z_1) (C-x)| ^m \mathrm d x \leq \mathrm{Leb}(B\cap C){\mathbb E} Z^m{\mathbb E} |M|^m .\end{equation}

Proof of Corollary 1. In order to apply Theorem 3 we need to verify (15) with $H(C-x,z)\;:\!=\;\upsilon(z)(C-x)$ , $C\in\mathcal{B}(\mathbb R^d)$ , $x\in\mathbb R^d$ , $z\in\textbf{Z}\;:\!=\;\textbf{N}_{\mathbb R^d\times{\mathbb R}}$ , and $\lambda(\cdot)\equiv\lambda \boldsymbol 1 _B(\cdot)$ . For the lower bound we note that

\begin{align*} \int_{B} \mathbb E \upsilon(Z_1) (C-x)^2 \mathrm d x=\int_{B}\mathbb E\left(\sum _{k=0}^{Z_1(C-x,{\mathbb R})-1}M_{1,k}\right)^2\mathrm d x \geq \int_{B\cap C}\mathbb E\left(\sum _{k=0}^{Z_1(C-x,{\mathbb R})-1}M_{1,k}\right)^2\mathrm d x. \end{align*}

Expanding the square of the sum and using the independence we have that

$$\mathbb E\left(\sum _{k=1}^{Z_1(C-x,{\mathbb R})}M_{1,k}\right)^2={\mathbb E}[Z_1(C-x,{\mathbb R})]{\mathbb E} M^2+{\mathbb E}[Z_1(C-x,{\mathbb R})(Z_1(C-x,{\mathbb R})-1)]({\mathbb E} M)^2\nonumber.$$

For $x\in B\cap C$ , we have $x\in C$ , and so the set $C-x$ contains the origin. Since $Z_1(\{\textbf{0}\},{\mathbb R})=1$ , we then have

$$\mathbb E\left(\sum _{k=1}^{Z_1(C-x,{\mathbb R})}M_{1,k}\right)^2 \geq {\mathbb E} M^2.$$

Therefore,

(21) \begin{align} \int_{B} \mathbb E \upsilon(Z_1) (C-x)^2 \mathrm d x & \geq\int_{B\cap C} \mathbb E \upsilon(Z_1) (C-x)^2 \mathrm d x\geq \mathrm{Leb}(B\cap C)\mathbb E M^2>0. \end{align}

As far as the upper bound is concerned, we note that by the bound on the Wasserstein distance in Theorem 3 and the inequalities (21) and (20), it immediately follows that

\begin{align*} d_W(F(C),G)\leq\frac{\lambda\int_{B}\mathbb{E}|\upsilon(Z_1)(C-x)|^3\mathrm{d}x}{\left(\int_{B}\lambda\mathbb{E}\upsilon(Z_1)(C-x)^2\mathrm{d}x\right)^{3/2}}&\leq \frac{\lambda \mathrm{Leb}(B\cap C)\mathbb E Z^3 {\mathbb E} |M|^3}{\left(\lambda{\mathbb E} M^2\mathrm{Leb}(B\cap C) \right)^{3/2}}\\[5pt] &=\frac{{\mathbb E} |M|^3\mathbb E Z^3}{({\mathbb E} M^2)^{3/2}\sqrt{\lambda \mathrm{Leb} (B\cap C)}}. \end{align*}

Similarly, the upper bound on the Kolmogorov distance follows from the upper bound on the Kolmogorov distance in Theorem 3, and again the inequalities (21) and (20).

6.2. Moderate deviations, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction term

In this section we apply Theorem 4 to the sequence $\{W_\ell(C_\ell)\}_{\ell\geq 1}$ , when the Poisson processes $\{X_n^{(\ell)}\}$ , $\ell\geq 1$ , have a piecewise deterministic intensity function and $\mathbb Q_\ell\equiv\mathbb Q$ for every $\ell\geq 1$ . In such a case the assumption (17) is greatly simplified. Moreover, the next corollary paves the way to the application of the theorem to some classes of generalized compound Hawkes processes (see Corollaries 4 and 6).

Corollary 2. Let $\{(B_\ell,C_\ell)\}_{\ell\in\mathbb N}\subset\mathcal{B}(\mathbb R^d)^2$ be such that $0<\mathrm{Leb}(B_\ell \cap C_\ell)<+\infty$ , $\ell\in\mathbb N$ . Assume that $\lambda_\ell(x)=\lambda_\ell \boldsymbol 1_{B_\ell}(x)$ , $x\in\mathbb R^d$ , for positive constants $\lambda_\ell>0$ , $\ell\in\mathbb N$ , $\mathbb Q_\ell\equiv \mathbb Q$ , ${\mathbb E} M^2>0$ , and

(22) \begin{equation}\frac{{\mathbb E} |M|^m{\mathbb E} Z^m}{({\mathbb E} M^2)^{m/2}\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}}\leq \frac{(m!)^{1+\gamma}}{\Delta_\ell ^{m-2}},\quad\text{for all $m\geq 3$ and $\ell\in\mathbb N$,}\end{equation}

for some $\gamma\geq 0$ and a positive numerical sequence $\{\Delta_\ell\}_{\ell \in \mathbb N}$ . Then the sequence $\{W_\ell(C_\ell)\}_{\ell\geq 1}$ satisfies an $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , a $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , and an $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ .

Proof. By (21), (20), the choice of the Borel sets $B_\ell$ and $C_\ell$ , $\ell\in\mathbb N$ , the assumption (22), and the fact that ${\mathbb E} M^2>0$ , we have

\begin{equation*}0<\mathrm{Leb}(B_\ell \cap C_\ell){\mathbb E} M^2\leq\int_{B_\ell} \mathbb E\upsilon(Z_1)(C_\ell-x)^2 \mathrm d x\end{equation*}

and

\begin{equation*}\int_{B_\ell} \mathbb E|\upsilon(Z_1)(C_\ell-x)|^m \mathrm d x \leq\mathrm{Leb}(B_\ell \cap C_\ell){\mathbb E} |M|^m\mathbb E Z^m<\infty,\quad\text{for every $m\geq 3$.}\end{equation*}

Therefore the condition (16) holds, and using again the assumption (22), we have

\begin{align*}\frac{\lambda_\ell\int_{B_\ell} {\mathbb E} |\upsilon(Z_1)(C_\ell-x)|^m \mathrm d x}{\left( \lambda_\ell\int_{B_\ell} {\mathbb E} (\upsilon(Z_1)(C_\ell-x))^2 \mathrm d x\right)^{\frac{m}{2}}}&\leq \frac{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell){\mathbb E} |M|^m\mathbb E Z^m}{(\lambda_\ell \mathrm {Leb}(B_\ell \cap C_\ell){\mathbb E} M^2)^{\frac{m}{2}} }\\[5pt] &= \frac{{\mathbb E} |M|^m\mathbb E Z^m}{({\mathbb E} M^2)^{m/2}\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}}\\[5pt] &\leq \frac{(m!)^{1+\gamma}}{\Delta_\ell ^{m-2}},\quad\text{for all $m\geq 3$ and $\ell\in\mathbb N$.}\end{align*}

The claim follows by Theorem 4.

7.1. Generalized compound Hawkes processes with Poisson offspring distribution

7.1.1. Gaussian approximation

In this paragraph we suppose that Z is distributed as the total progeny of a Galton–Watson process with one ancestor and offspring distribution the Poisson law with mean $h\in (0,1)$ , and that $\{X_n\}_{n\geq 1}$ is a Poisson process on ${\mathbb R}^d$ with intensity function $\lambda(x)=\lambda\textbf{1}_B(x)$ , $x\in\mathbb R^d$ , for some $\lambda>0$ and some Borel set $B\subseteq\mathbb R^d$ . We denote by $V_{\mathrm{Poisson}}$ the corresponding generalized compound Hawkes process and by $W_{\mathrm{Poisson}}$ the functional (8) with $V_{\mathrm{Poisson}}$ in place of V.

Corollary 3. Under the foregoing assumptions and notation, if the Borel sets B and C are such that $0<\mathrm{Leb}(B\cap C)<+\infty$ and ${\mathbb E} M^2\in (0,\infty)$ , then the bounds (18) and (19) hold with $W_{\mathrm{Poisson}}$ in place of W,

(32) \begin{equation}\mathbb E Z^3=\frac{1+2h}{(1-h)^5},\end{equation}

and

(33) \begin{align} \mathbb E Z^4=\frac{1}{1-h} \Bigg [1+ \frac{4h}{1-h} +\frac{6h^2}{(1-h)^2}+\frac{4h^3+6h}{(1-h)^3} + \frac{h^4+12h^2}{(1-h)^4 } +\frac{6h^3}{(1-h)^5}+\frac{11h^2+4h}{(1-h)^6}\Bigg].\end{align}

Proof. Note that the conditions (23) and (24) are satisfied (the latter holds for any $n\in\mathbb N$ ). Therefore, the expressions for the third and fourth moments of the total progeny are given by Proposition 1. The claim follows by Corollary 1.

7.1.2. Moderate deviations, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction term

In this paragraph we suppose that, for each $\ell\in\mathbb N$ , $Z=Z_1^{\ell}({\mathbb R}^d,{\mathbb R})$ is distributed as the total progeny of a Galton–Watson process with one ancestor and offspring distribution the Poisson law with mean $h\in (0,1)$ , and that $\{X_n^{(\ell)}\}_{n\geq 1}$ is a Poisson process on ${\mathbb R}^d$ with intensity function $\lambda_\ell (x)=\lambda_\ell \boldsymbol 1_{B_\ell}(x)$ , $x\in\mathbb R^d$ , for positive constants $\lambda_\ell>0$ and Borel sets $B_\ell\subseteq\mathbb{R}^d$ , $\ell \in \mathbb N$ . We denote by $V_{\mathrm{Poisson}}^{(\ell)}$ the corresponding generalized compound Hawkes process and by $W_{\mathrm{Poisson}}^{(\ell)}$ the functional (10) with $V_{\mathrm{Poisson}}^{(\ell)}$ in place of $V_\ell$ .

Corollary 4. Let the foregoing assumptions and notation prevail, and let the Borel sets $B_\ell$ and $C_\ell$ , $\ell \in \mathbb N$ , be such that $0<\mathrm{Leb}(B_\ell \cap C_\ell)<+\infty$ , $\ell\in\mathbb N$ , ${\mathbb E} M^2>0$ , and

(34) \begin{equation}\frac{{\mathbb E} |M|^m}{({\mathbb E} M^2)^{m/2}}\leq (m!)^{\gamma},\quad\text{for any $m\geq 3$ and some $\gamma\geq 0$.}\end{equation}

Then the following hold:

1. (i) If $h-1-\log h\geq 1$ , then the sequence $\{W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\}_{\ell\geq 1}$ satisfies $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , where

   \[\Delta_\ell\;:\!=\;h\sqrt{\lambda_\ell\text{Leb}(B_\ell \cap C_\ell)}.\]

2. (ii) If $h-1-\log h<1$ , then the sequence $\{W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\}_{\ell\geq 1}$ satisfies $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , where

   \[\Delta_\ell \;:\!=\; h(h-1-\log h)^3\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}.\]

Example 1. Note that the condition (34) is trivially satisfied with $\gamma=0$ if M is a constant different from zero. Similarly, if M is a uniform variable on [0, D] with $D> 0$ , the moments of M satisfy

$$\frac{{\mathbb E} |M|^m}{({\mathbb E} M^2)^{m/2}}=\frac{3^{\frac{m}{2}}}{m+1}<m!.$$

Assumption (34) holds with $\gamma=1$ even if M is exponentially distributed with mean $\mu^{-1}$ , for some $\mu>0$ . Indeed, in such a case we have

\[\frac{{\mathbb E} |M|^m}{({\mathbb E} M^2)^{m/2}}=\frac{m!}{2^{m/2}}<m!.\]

Another example is M Gaussian distributed with mean zero and variance $\sigma^2$ . Indeed, in such a case we have

$$\frac{{\mathbb E} |M|^m}{({\mathbb E} M^2)^{m/2}} \leq (m-1)!!.$$

By distinguishing the two cases $m=2p+1$ and $m=2p+2$ for $p=2,3,\cdots$ , we conclude that

$$\frac{{\mathbb E} |M|^m}{({\mathbb E} M^2)^{m/2}}\leq \sqrt {m!},$$

which implies that the condition (34) is satisfied with $\gamma=\frac{1}{2}.$

The proof of the corollary exploits the following lemmas, which are proved in Appendix A.

Lemma 3. For any $\nu>0$ and any integer $m\geq 2$ we have

$$\sum_{k=1}^{+\infty}\mathrm{e}^{-\nu k}k^{m-1}=\nu ^{-m} (m-1)!+ R_m,$$

with

$$|R_m| \leq \frac{1}{\pi(m-1)}+ \frac{2(m-1)!}{\pi^m} .$$

Lemma 4. The function

\[f(x)\;:\!=\;x(x-1-\log x)^2,\quad x\in (0,1),\]

is such that $f(x)\in (0,1)$ .

Proof of Corollary 4. It is well-known that the total progeny Z of a sub-critical Galton–Watson process with one ancestor and Poisson offspring law with mean $h\in (0,1)$ follows the Borel distribution (cf. [Reference Privault37] and the references therein), i.e.,

$$\mathbb P (Z=k)=\frac{\mathrm e^{-hk}(hk)^{k-1}}{k!}, \quad k=1,2,\ldots.$$

Therefore, by Stirling’s inequality, for $m\geq 3$ we have

\begin{align*} {\mathbb E} Z^m = \sum_{k=1}^{+\infty} \frac{\mathrm e^{-hk}(hk)^{k-1}}{k!} k^m &\leq \sum_{k=1}^{+\infty}\mathrm e^{-hk}(hk)^{k-1} k^m \left(\frac{\mathrm e}{k} \right)^k \frac{1}{\sqrt {2\pi k}}\\[5pt] &= \frac{1}{\sqrt {2\pi}} \sum_{k=1}^{+\infty} \mathrm e^{(1-h)k} \frac{k^m}{k\sqrt k} h^{k-1}\\[5pt] &= \frac{1}{h\sqrt {2\pi }} \sum_{k=1}^{+\infty} \mathrm e^{-(h-1-\log h)k} k^{m-1.5} \\[5pt] &\leq \frac{1}{h\sqrt {2\pi }} \sum_{k=1}^{+\infty} \mathrm e^{-(h-1-\log h)k} k^{m-1}.\end{align*}

Using Lemma 3 with $\nu\;:\!=\;h-1-\log h>0$ , we have

(35) \begin{equation}{\mathbb E} Z^m \leq \frac{(m-1)!}{h\sqrt{2\pi}}\left(\nu^{-m}+\frac{2}{\pi^m}+ \frac{1}{\pi (m-1)(m-1)!}\right),\quad\text{$m\geq 3$.}\end{equation}

We now give separately the proofs of Parts (i) and (ii); in both cases we shall apply Corollary 2.

Proof of Part (i).

If $\nu\geq 1$ , then, for any $m\geq 3$ ,

(36) \begin{equation}\nu^{-m}+\frac{2}{\pi^m}+ \frac{1}{\pi (m-1)(m-1)!} \leq 3\leq m.\end{equation}

Combining this inequality with (35), for $m\geq 3$ and $\ell\in\mathbb N$ , we have

\begin{align*}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2} }\leq& \frac{1}{h\sqrt{2\pi}}\frac{m!}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}}\\[5pt] =& \frac{h^{m-3}}{\sqrt{2\pi}}\frac{m!}{(h\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)})^{m-2}}\\[5pt] \leq & \frac{m!}{(h\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)})^{m-2}}\end{align*}

since $h<1$ and $\sqrt{2\pi}>1$ . Combining this latter inequality with the assumption (34) we have that the condition (22) is satisfied with $\Delta_\ell=h\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}$ , and the claim follows by Corollary 2.

Proof of Part (ii).

Now we suppose $\nu < 1$ . Since

(37) \begin{equation}\frac{2}{\pi^m}+\frac{1}{\pi (m-1)(m-1)!}\leq\frac{2}{\pi}\left(\frac{1}{\pi^2}+\frac{1}{8}\right)<0.16,\quad\text{for any $m\geq 3$,}\end{equation}

by (35) we have

\begin{align*}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}} \leq & \frac{1}{h\sqrt{2\pi}}\frac{(m-1)!}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}}\left(\frac{1}{\nu^{m}}+0.16\right)\\[5pt] = &\frac{(m-1)!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu)^{m-2}}\frac{\nu^m}{h\nu^2\sqrt{2\pi}}\left(\frac{1}{\nu^{m}}+0.16\right),\quad\text{$m\geq 3$, $\ell\in\mathbb N$}.\end{align*}

Since $\nu <1$ , we have

\begin{align*} \frac{\nu^m}{h \nu^2\sqrt{2\pi} }\left(\frac{1}{\nu^{m}}+0.16\right) &\leq\frac{1.16}{h\nu^2\sqrt{2\pi}}<\frac{1}{h\nu^2}\;:\!=\;u_h.\end{align*}

Therefore

\begin{align*}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell )}^{m-2}} \leq &\frac{(m-1)!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu)^{m-2}}u_h\\[5pt] =&\frac{(m-1)!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu u_h^{-1})^{m-2}}u_h^{3-m},\quad m\geq 3, \ell\in\mathbb N.\end{align*}

By Lemma 4 we have $u_h^{-1} <1$ . Therefore, for any $m\geq 3$ and $\ell\in\mathbb N$ , we have

\begin{align*}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}} \leq & \frac{m!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu u_h^{-1})^{m-2}}.\end{align*}

Combining this latter inequality with the assumption (34) we have that the condition (22) is satisfied with $\Delta_\ell \;:\!=\; h\nu^3\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}$ , and the claim follows by Corollary 2.

7.2. On the Gaussian approximation bound in the Kolmogorov distance and the normal approximation with Cramér correction term

The aim of this section is to illustrate, by means of a simple example, the differences and the analogies between Gaussian approximation bounds in the Kolmogorov distance and normal approximations with Cramér correction term.

Let $W_{\mathrm{Poisson}}^{(\ell)}$ , $\ell\in\mathbb N$ , be defined as at the beginning of Section 7.1.2, with $\{(B_\ell,C_\ell)\}_{\ell\in\mathbb N}\subset\mathcal{B}(\mathbb{R}^d)^2$ a sequence of Borel sets such that $0<\mathrm{Leb}(B_\ell\cap C_\ell)<\infty$ , $\ell\in\mathbb N$ , and $\lambda_\ell\mathrm{Leb}(B_\ell\cap C_\ell)\to+\infty$ as $\ell\to+\infty$ . Assume $M\equiv 1$ (so that (34) holds with $\gamma=0$ ), and let $\Delta_\ell$ be defined as in Part (i) or (ii) of Corollary 4. We know that $\{W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\}_{\ell\geq 1}$ satisfies $\textbf{NACC}(0,\{\Delta_\ell\}_{\ell\in\mathbb N})$ .

For a fixed $0<r<1/3$ , let $\ell^*$ be sufficiently large so that

\[0<\Delta_{\ell}^r<c_0\Delta_{\ell},\quad\text{for all $\ell\geq\ell^*$,}\]

where $c_0$ is the positive constant which appears in the definition of $\textbf{NACC}(0,\{\Delta_\ell\}_{\ell\in\mathbb N})$ . Setting $x_\ell\;:\!=\;\Delta_{\ell}^r$ , for all $\ell\geq\ell^*$ , we have

\begin{align}\mathbb{P}(W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\geq x_\ell)-\mathbb{P}(G\geq x_\ell)&=\mathrm{e}^{L_{\ell,x_\ell}^+}\mathbb P(G\geq x_\ell)\left(1+c_1\theta_{\ell,x_\ell}^+\frac{1+x_\ell}{\Delta_{\ell}}\right)-\mathbb{P}(G\geq x_\ell)\nonumber\\[5pt] &=(\mathrm{e}^{L_{\ell,x_\ell}^+}-1)\mathbb{P}(G\geq x_\ell)+c_1\theta_{\ell,x_\ell}^{+}\mathrm{e}^{L_{\ell,x_\ell}^+}\mathbb P(G\geq x_\ell)\frac{1+x_\ell}{\Delta_{\ell}}.\nonumber\end{align}

Since $|L_{\ell,x_\ell}^{+}|\leq c_2\Delta_\ell^{3r-1}$ and $|\theta_{\ell,x_\ell}^+|\leq 1$ , we have

\begin{align}|\mathbb{P}(W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\geq x_\ell)-\mathbb{P}(G\geq x_\ell)|&\leq \left(c_2\Big|\frac{\mathrm{e}^{L_{\ell,x_\ell}^+}-1}{L_{\ell,x_\ell}^+}\Big|\Delta_\ell^{3r-1}+c_1 O(1)\Big|\frac{1+x_\ell}{\Delta_{\ell}}\Big|\right) \mathbb P(G\geq x_\ell)\nonumber\\[5pt] &=\left(c_2 O(1)\Delta_\ell^{3r-1}+c_1 O(1)\Big|\frac{1+\Delta^r_\ell}{\Delta_{\ell}}\Big|\right) \mathbb P(G\geq x_\ell)\nonumber\\[5pt] &=O(\Delta_\ell^{3r-1})\mathbb P(G\geq x_\ell).\nonumber\end{align}

Bounding the Gaussian tail from above, we obtain

(38) \begin{align} |\mathbb{P}(W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\geq x_\ell)-\mathbb{P}(G\geq x_\ell)|&\leq O(\Delta_\ell^{3r-1})\frac{\mathrm{e}^{-x_\ell^2/2}}{x_\ell \sqrt{2\pi}}\nonumber\\[5pt] &=O(\Delta_\ell^{2r-1}\mathrm{e}^{-\Delta_\ell ^{2r}/2})\nonumber\\[5pt] &=O\left(\frac{\exp\left({-\frac{u_h^{2r}(\lambda_\ell\mathrm{Leb}(B_\ell\cap C_\ell))^r}{2}}\right)}{(\lambda_\ell\mathrm{Leb}(B_\ell\cap C_\ell))^{\frac{1}{2}-r}} \right),\quad\text{as $\ell\to+\infty$,}\end{align}

where either $u_h\;:\!=\;h$ or $u_h\;:\!=\;h(h-1-\log h)^3$ . On the other hand, the bound (19) and the relations (32) and (33) yield

(39) \begin{align}|\mathbb{P}(W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\geq x_\ell)-\mathbb{P}(G\geq x_\ell)|=O\left(\frac{1}{(\lambda_\ell\mathrm{Leb}(B_\ell\cap C_\ell))^{1/2}}\right),\quad\text{as $\ell \to +\infty$ .}\end{align}

Clearly the rate (38) is much faster than that of (39). Now let $x\in (0,\infty)$ be arbitrarily fixed. For $\ell$ large enough we have $x\in [0,c_0\Delta_\ell]$ , and so by Corollary 4, for all $\ell$ large enough, we have

\begin{align}|\mathbb{P}(W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\geq x)-\mathbb{P}(G\geq x)|&\leq |\mathrm{e}^{L_{\ell,x}^+}-1|\mathbb{P}(G\geq x)+c_1\mathrm{e}^{L_{\ell,x}^+}\mathbb P(G\geq x)\theta_{\ell,x}^+\frac{1+x}{\Delta_\ell}\nonumber\\[5pt] &\leq c_2\Big|\frac{\mathrm{e}^{L_{\ell,x}^+}-1}{L_{\ell,x}^+}\Big|\frac{x^3}{\Delta_\ell}\mathbb{P}(G\geq x)+c_1\mathrm{e}^{L_{\ell,x}^+}\mathbb P(G\geq x)\theta_{\ell,x}^+\frac{1+x}{\Delta_\ell}\nonumber\\[5pt] &=O\left(\frac{1}{\sqrt{\lambda_\ell\mathrm{Leb}(B_\ell\cap C_\ell)}}\right).\nonumber\end{align}

Clearly, the same rate is provided by the bound (19) and the relations (32) and (33). We emphasize that (i) the inequality (19) and the relations (32) and (33) do indeed yield an explicit bound on the quantity $|\mathbb{P}(W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\geq x)-\mathbb{P}(G\geq x)|$ , for any $x\in\mathbb R$ and any $\ell\in\mathbb N$ ; (ii) an explicit bound on the quantity $|\mathbb{P}(W_{\mathrm{Poisson}}^{(\ell)}(C_\ell)\geq x)-\mathbb{P}(G\geq x)|$ , for any $x\in\mathbb R$ and any $\ell\in\mathbb N$ , is not amenable via the normal approximation with Cramér correction term (for various obvious reasons).

7.3. Comparison with some related literature

7.3.1. Gaussian approximation

Let N be a classical Hawkes process on $(0,\infty)$ with parameters $(\lambda,g)$ . Then Corollary 3 with $B=(0,\infty)$ and $C=(0,t]$ , $t>0$ , gives explicit bounds for the Gaussian approximation of

\[\frac{N((0,t])-{\mathbb E} N((0,t])}{\sqrt{\mathbb V\mathrm{ar}(N((0,t]))}},\]

both in the Wasserstein and in the Kolmogorov distance. Note that for the fertility function g we assume only the standard stability condition $h\;:\!=\;\int_0^\infty g(t)\,\mathrm{d}t\in (0,1)$ .

It is worthwhile to compare these bounds with the ones in [Reference Hillairet, Huang, Khabou and Réveillac23, Reference Khabou, Privault and Réveillac26]. Theorem 3.13 in [Reference Hillairet, Huang, Khabou and Réveillac23] gives a bound of the kind

\[d_W\left(\frac{N((0,t])-\mathbb E N((0,t])}{\sqrt{t}},G\right)\leq c/\sqrt t, \quad t>0,\]

for some constant $c>0$ which is not explicitly computed and for specific choices of g (exponential and Erlang). This result has been extended in [Reference Khabou, Privault and Réveillac26] to fertility functions $g\;:\; [0,\infty)\to [0,\infty)$ such that $h\in (0,1)$ and $\int_0^\infty t g(t)\mathrm{d}t<\infty$ . The techniques used in [Reference Hillairet, Huang, Khabou and Réveillac23, Reference Khabou, Privault and Réveillac26] are based on the Poisson embedding construction of Hawkes processes and the Malliavin calculus on the Poisson space. These ideas were previously used in [Reference Torrisi44, Reference Torrisi45] for the purpose of Gaussian and Poisson approximation of some classes of nonlinear Hawkes processes. Note that, in contrast with classical Hawkes processes, nonlinear Hawkes processes (introduced in [Reference Brémaud and Massoulié8]) do not have a Poisson cluster representation. For Poisson cluster processes (such as classical Hawkes processes), the number of points on some measurable set can be represented as an integral with respect to a suitable Poisson random measure. As a consequence, results on the Gaussian approximation of the number of points on a measurable set can be obtained by applying the general results in [Reference Last, Peccati and Schulte30].

7.3.2. Moderate deviations

In this section we compare Corollary 4 with a couple of related results in the literature. Firstly, we prove that Corollary 4, when specialized to a classical Hawkes process on $(0,\infty)$ , implies the same moderate deviation principle provided in [Reference Zhu47] (see Theorem 1 therein), with an alternative assumption on the fertility function of the process. We refer the reader to [Reference Gao and Wang19] for sample-path moderate deviation principles, on the space of càdlàg functions on [0, 1] equipped with the Skorokhod topology, for Poisson cluster point processes on the line; see [Reference Gao and Wang19, Theorem 2]. Secondly, we compare Corollary 4 (again specialized to a classical Hawkes process on $(0,\infty)$ ) with Theorem 8 in [Reference Gao and Zhu20]. Hereafter, N denotes a classical Hawkes process on $(0,\infty)$ with parameters $(\lambda,g)$ , and we assume that g satisfies the usual stability condition $h\;:\!=\;\int_0^\infty g(t)\,\mathrm{d}t\in (0,1)$ .

Corollary 4 with $\lambda_\ell=\lambda>0$ , $B_\ell=(0,\infty)$ , and $C_\ell=(0,\ell]$ , $\ell\in\mathbb N$ , and $M\equiv 1$ yields that, for any sequence of positive numbers $\{a_\ell\}_{\ell\in\mathbb N}$ such that $\lim_{\ell\to\infty}a_\ell=+\infty$ and $\lim_{\ell\to\infty}\frac{a_{\ell}}{\sqrt{\ell}}=0$ , for any Borel set $B\subseteq{\mathbb R}$ ,

\begin{align}-\inf_{x\in\overset{\circ}B}\frac{x^2}{2}&\leq\liminf_{\ell\to\infty}a_\ell^{-2}\log\mathbb{P}\Biggl(\frac{N((0,\ell])-\mathbb E N((0,\ell])}{a_\ell\sqrt{\mathbb{V}\mathrm{ar}(N((0,\ell]))}}\in B\Biggr)\nonumber\\[5pt] &\leq\limsup_{\ell\to\infty}a_\ell^{-2}\log\mathbb{P}\Biggl(\frac{N((0,\ell])-\mathbb E N((0,\ell])}{a_\ell\sqrt{\mathbb{V}\mathrm{ar}(N((0,\ell]))}}\in B\Biggr)\leq-\inf_{x\in\overline B}\frac{x^2}{2}.\nonumber\end{align}

Reasoning by contradiction, one has that, for any function $a(\cdot)$ such that $\lim_{t\to\infty}a(t)=+\infty$ and $a(t)=o(\sqrt t)$ , as $t\to+\infty$ , and any Borel set $B\subseteq\mathbb R$ ,

\begin{align}-\inf_{x\in\overset{\circ}B}\frac{x^2}{2}&\leq\liminf_{t\to\infty}a(t)^{-2}\log\mathbb{P}\Biggl(\frac{N((0,t])-\mathbb E N((0,t])}{a(t)\sqrt{\mathbb{V}\mathrm{ar}(N((0,t]))}}\in B\Biggr)\nonumber\\[5pt] &\leq\limsup_{\ell\to\infty}a(t)^{-2}\log\mathbb{P}\Biggl(\frac{N((0,t])-\mathbb E N((0,t])}{a(t)\sqrt{\mathbb{V}\mathrm{ar}(N((0,t]))}}\in B\Biggr)\leq-\inf_{x\in\overline B}\frac{x^2}{2}.\nonumber\end{align}

It is easily realized (cf. [Reference Bacry, Delattre, Hoffmann and Muzy4] for example) that, under the stability condition,

\[\mathbb E N((0,t])/t\to\frac{\lambda}{1-h}\quad\text{and}\quad\mathbb{V}\mathrm{ar}(N((0,t]))/t\to\frac{\lambda}{(1-h)^3},\quad\text{as $t\to+\infty$.}\]

So, letting $b(\cdot)$ denote a function such that $\sqrt t=o(b(t))$ and $b(t)=o(t)$ , as $t\to\infty$ , and setting $a(t)\;:\!=\;b(t)/\sqrt{\mathbb{V}\mathrm{ar}(N((0,t]))}$ , we have that, for any Borel set $B\subseteq\mathbb R$ ,

(40) \begin{align}-\inf_{x\in\overset{\circ}B}\frac{x^2(1-h)^3}{2\lambda}&\leq\liminf_{t\to\infty}\frac{t}{b(t)^2}\log\mathbb{P}\Biggl(\frac{N((0,t])-\mathbb E N((0,t])}{b(t)}\in B\Biggr)\nonumber\\[5pt] &\leq\limsup_{\ell\to\infty}\frac{t}{b(t)^2}\log\mathbb{P}\Biggl(\frac{N((0,t])-\mathbb E N((0,t])}{b(t)}\in B\Biggr)\leq-\inf_{x\in\overline B}\frac{x^2(1-h)^3}{2\lambda}.\end{align}

By Lemma 5 in [Reference Bacry, Delattre, Hoffmann and Muzy4], we have that, if in addition to the stability condition $h\in (0,1)$ we assume

(41) \begin{equation}\int_0^\infty\sqrt t g(t)\,\mathrm{d}t<\infty,\end{equation}

then

\[\frac{\mathbb E N((0,t])-\lambda t/(1-h)}{b(t)}\to 0,\quad\text{as $t\to\infty$.}\]

So, for an arbitrarily fixed $\delta>0$ , there exists $t_\delta$ such that for any $t>t_\delta$ it holds that

\[\Big|\frac{N((0,t])-\mathbb E N((0,t])}{b(t)}-\frac{N((0,t])-\frac{\lambda t}{1-h}}{b(t)}\Big|=\Big|\frac{\mathbb E N((0,t])-\lambda t/(1-h)}{b(t)}\Big|<\delta.\]

Therefore, for an arbitrarily fixed $\delta>0$ , there exists $t_\delta$ such that for any $t>t_\delta$ we have

\[\log\mathbb{P}\Biggl(\Big|\frac{N((0,t])-\mathbb E N((0,t])}{b(t)}-\frac{N((0,t])-\frac{\lambda t}{1-h}}{b(t)}\Big|>\delta\Biggr)=-\infty.\]

Hence the processes $\{\frac{N((0,t])-\mathbb E N((0,t])}{b(t)}\}_{t>0}$ and $\{\frac{N((0,t])-\lambda t/(1-h)}{b(t)}\}_{t>0}$ are exponentially equivalent (see [Reference Dembo and Zeitouni12, Definition 4.2.10, p. 130]). Therefore, by [Reference Dembo and Zeitouni12, Theorem 4.2.13, p. 130], the relation (40) holds with $\mathbb E N((0,t])$ replaced by $\lambda t/(1-h)$ . Thus we recover the moderate deviation principle proved in [Reference Zhu47] under an alternative condition on g (the latter paper assumes the stability condition and $\sup_{t>0}t^{3/2}g(t)<\infty$ , which is clearly different from (41)).

Theorem 8 in [Reference Gao and Zhu20] states that, under the assumption

$$\int_0 ^{\infty} t g(t) {\mathrm d} t <+\infty$$

(which is clearly stronger than (41)), for any $y(t)=o(t^{1/2-1/m})$ as $t\to+\infty$ , any integer $m\geq 3$ , and any positive function $b(\cdot)$ , it holds that

\begin{equation*} \mathbb P \left (\frac{N((0,t])-\frac{\lambda t}{1-h}}{b(t)} \geq \frac{\sqrt{t}y(t)}{b(t)} \frac{\sqrt{\lambda}}{(1-h)^{3/2}} \right)=\frac{(1+o(1))}{y(t) \sqrt{2\pi}} \mathrm e ^{-\sum_{i=2}^{m-1}c_i \frac{y(t)^i}{t^{(i-2)/2}}},\quad\text{as $t\to+\infty$,}\end{equation*}

where $\{c_i\}_{i=1,\cdots,m-2}$ are real coefficients that can be computed explicitly; for instance, one has $c_2=\frac 12 $ . In particular, if $b(t)=o( t^{2/3})$ , as $t\to+\infty$ , then by choosing

\[y(t)=K \frac{(1-h)^{3/2}}{\sqrt {\lambda}}\frac{b(t)}{\sqrt{t}} =o(t^{1/2-1/3}),\quad\text{as $t\to+\infty$, for some $K>0$,}\]

we have

(42) \begin{equation} \mathbb P \left( \frac{N((0,t])-\frac{\lambda t}{1-h}}{b(t)} \geq K\right)= \frac{\sqrt \lambda (1+o(1)) }{(1-h)^{3/2}K}\frac{\sqrt t}{b(t)} \mathrm e ^{-\frac{K^2}{2} \frac{(1-h)^{3}}{\lambda}\frac{b(t)^2}{t}},\end{equation}

which is a more precise form of the relation (40) for the Borel set $B=[K,+\infty)$ . Note that, unlike the formula (40), which is valid for any Borel set B, the formula (42) gives asymptotic estimates only for half-lines.

7.3.3. Bernstein-type concentration inequalities

In this section we present some consequences of Corollary 4 concerning stationary compound Hawkes processes on the line, i.e., $B_T\;:\!=\;{\mathbb R}$ , observed on the time interval $C_T\;:\!=\;(0,T]$ ; here T replaces $\ell$ to emphasize the dependence on time.

If we interpret the mark M as the claim that an insurer must pay to an insurance policy holder, then the variable V((0, T]) (defined by (3)) represents the total loss incurred by the insurer in the time interval (0, T].

Assume that the claims arrivals are modeled by the points of a Hawkes process of baseline intensity $\lambda>0$ and Poisson offspring distribution with mean $h\in (0,1)$ satisfying $h-1-\log h \geq 1$ . Assume moreover that the mark M follows the exponential distribution of parameter $\mu^{-1} \in (0,+\infty)$ . Then Corollary 4 yields

\[\mathbb P \left( \left | \frac{V((0,T])-{\mathbb E} V((0,T])}{\sqrt{\mathbb{V}\mathrm{ar} V((0,T])}}\right | \geq x \right) \leq 2 \exp \left (-\frac{1}{4} \min \left \{\frac{x^2}{2^{1+\gamma}}, (x\Delta _T)^{\frac{1}{1+\gamma}} \right \}\right),\quad\text{$x\geq 0$},\]

where $\Delta_T=h\sqrt{\lambda T}$ and $\gamma=1$ by virtue of Example 1. By stationarity, this inequality can be rewritten as

\begin{align}&\mathbb P \left (V((0,T]) \in \left[\frac{\lambda \mu}{1-h} T-x \sqrt{\frac{2\mu^2 \lambda}{(1-h)^3}T }, \frac{\lambda \mu}{1-h} T+x \sqrt{\frac{2\mu^2 \lambda}{(1-h)^3}T } \right]\right)\nonumber\\[5pt] &\qquad\qquad\geq 1-2 \exp \left (-\frac{1}{4} \min \left \{\frac{x^2}{4}, (xh\sqrt{\lambda T})^{1/2} \right \}\right),\quad\text{$x\geq 0$},\nonumber\end{align}

which yields a non-asymptotic lower bound on the probability that the total loss is within x times its standard deviation.

Another quantity of interest for insurers is the probability that the total loss greatly exceeds its expected value. The Bernstein-type concentration inequality, being valid for any $x\geq 0$ , yields an upper bound on this probability. Indeed, by choosing

\[x=x_T=(k-1)\sqrt{\frac{\lambda(1-h)T}{2}},\quad\text{for some $k>1$,}\]

we have

(43) \begin{align} \mathbb P \left(V((0,T]) \geq k \frac{\lambda \mu}{1-h} T \right)&= \mathbb P \left (\frac{V((0,T]) - {\mathbb E} V((0,T])}{\sqrt{\mathbb{V}\mathrm{ar} V((0,T])}} \geq x_T\right)\nonumber\\[5pt] &\leq \mathbb P \left ( \left |\frac{V((0,T]) - {\mathbb E} V((0,T])}{\sqrt{\mathbb{V}\mathrm{ar} V((0,T])}}\right | \geq x_T\right)\nonumber\\[5pt] &\leq 2 \exp\!\left (-\frac{1}{4} \min\! \left \{\frac{(k-1)^2\lambda (1-h)T}{8}, \!\left(\!(k-1)\lambda h \sqrt{\frac{1-h}{2}} T\!\right)^{\!1/2}\!\right \} \!\right)\!. \end{align}

If the time horizon T satisfies $T \geq \frac{2^{11/2}h}{(k-1)^3 \lambda (1-h)^{3/2}}$ , then the inequality (43) simplifies to

(44) \begin{equation} \mathbb P \left(V((0,T]) \geq k \frac{\lambda \mu}{1-h} T \right) \leq 2\exp \left (-\frac{1}{4} \left((k-1)\lambda h \sqrt{\frac{1-h}{2}} T\right)^{1/2} \right). \end{equation}

A similar (non-asymptotic) inequality appears in Proposition 2.1 of [Reference Reynaud-Bouret and Roy39], albeit working only for stationary Hawkes processes on the line whose fertility functions have compact support, and involving quantities that are not explicitly known. We also point out that, by specializing the inequality (44) to the simple Hawkes process, that is, with constant marks ( $\gamma=0$ by virtue of Example 1), we can find a decay rate for the tail probability similar to the one given in [Reference Reynaud-Bouret and Roy39], but with explicit constants.

7.4. Generalized compound Hawkes processes with binomial offspring distribution

7.4.1. Gaussian approximation

In this paragraph we suppose that Z is distributed as the total progeny of a Galton–Watson process with one ancestor and offspring distribution the binomial law with parameters (h, p), with $h\in\mathbb N$ and $p\in (0,1)$ such that $hp\in (0,1)$ . We assume that $\{X_n\}_{n\geq 1}$ is a Poisson process on ${\mathbb R}^d$ with intensity function $\lambda(x)=\lambda\textbf{1}_B(x)$ , $x\in\mathbb R^d$ , for some positive constant $\lambda>0$ and some Borel set $B\subseteq\mathbb{R}^d$ . We denote by $V_{\mathrm{binomial}}$ the corresponding generalized compound Hawkes process and by $W_{\mathrm{binomial}}$ the functional (8) with $V_{\mathrm{binomial}}$ in place of V.

Corollary 5. Under the foregoing assumptions and notation, if the Borel sets B and C are such that $0<\mathrm{Leb}(B\cap C)<+\infty$ and ${\mathbb E} M^2\in (0,\infty)$ , then the bounds (18) and (19) hold with $W_{\mathrm{binomial}}$ in place of W,

\begin{align}&\mathbb E Z^3=\frac{1}{1-hp}\Biggl(1+\frac{3hp}{1-hp}+3\frac{p^2(h)_2}{(1-hp)^2}+\frac{p^3(h)_3+3hp(1-p)}{(1-hp)^3}+3\frac{h^2p^2(1-p)^2}{(1-hp)^4}\Biggr),\nonumber\end{align}

and

\begin{align*} \mathbb E Z^4=&\frac{1}{1-hp} \Bigg [1+\frac{4hp}{1-hp}+\frac{6p^2(h)_2}{(1-hp)^2}+4\frac{p^3(h)_3}{(1-hp)^3} + \frac{p^4(h)_4}{(1-hp)^4 }\\[5pt] &+\frac{3(1-hp^2)}{(1-hp)^3}\Big(2hp+\frac{4p^2(h)_2}{1-hp}+p^2(h)_2\frac{1-hp^2}{(1-hp)^3}+\frac{2p^3(h)_3}{(1-hp)^2}\Big)+4\mathbb E Z^3 \frac{hp(1-p)}{1-hp}\Bigg].\end{align*}

Here $(h)_n\;:\!=\;h(h-1)\ldots (h-(n-1))\textbf{1}_{\{h\geq n\}}$ .

Proof. Similar to the proof of Corollary 3.

7.4.2. Moderate deviations, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction term

In this paragraph we suppose that, for each $\ell\in\mathbb N$ , $Z=Z_1^{\ell}({\mathbb R}^d,{\mathbb R})$ is distributed as the total progeny of a Galton–Watson process with one ancestor and offspring distribution the binomial law with parameters (h, p), with $h\in\mathbb N$ and $p\in (0,1)$ such that $hp\in (0,1)$ . We assume that $\{X_n^{(\ell)}\}_{n\geq 1}$ is a Poisson process on ${\mathbb R}^d$ with intensity function $\lambda_\ell (x)=\lambda_\ell \boldsymbol 1_{B_\ell}(x)$ , $x\in\mathbb R^d$ , for positive constants $\lambda_\ell>0$ and Borel sets $B_\ell\subseteq\mathbb R^d$ , $\ell \in \mathbb N$ . We denote by $V_{\mathrm{binomial}}^{(\ell)}$ the corresponding generalized compound Hawkes process and by $W_{\mathrm{binomial}}^{(\ell)}$ the functional (10) with $V_{\mathrm{binomial}}^{(\ell)}$ in place of $V_\ell$ .

Corollary 6. Let the foregoing assumptions and notation prevail, and let the Borel sets $B_\ell$ and $C_\ell$ , $\ell \in \mathbb N$ , be such that $0<\mathrm{Leb}(B_\ell \cap C_\ell)<+\infty$ , $\ell\in\mathbb N$ , ${\mathbb E} M^2>0$ . Assume (34). Then the following hold:

1. (i.1) If $h=1$ and $p\leq\mathrm{e}^{-1}$ , then the sequence $\{W_{\mathrm{binomial}}^{(\ell)}(C_\ell)\}_{\ell\geq 1}$ satisfies $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , where

   \[\Delta_\ell\;:\!=\; \frac{p}{1.05(1-p)}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}.\]

2. (i.2) If $h=1$ and $p>\mathrm{e}^{-1}$ , then the sequence $\{W_{\mathrm{binomial}}^{(\ell)}(C_\ell)\}_{\ell\geq 1}$ satisfies $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , where

   \[\Delta_\ell\;:\!=\;\frac{p(\log p)^4}{1.05(1-p)}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}.\]

3. (ii.1) If $h\geq 2$ and $ph(h(1-p)/(h-1))^{h-1}\leq\mathrm{e}^{-1}$ , then the sequence $\{W_{\mathrm{binomial}}^{(\ell)}(C_\ell)\}_{\ell\geq 1}$ satisfies $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , where

   \[\Delta_\ell\;:\!=\;\left(1+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm e^{\frac{1}{24\cdot25}}(1-p)}{p(h-1)\sqrt {2\pi}}\right)^{-1}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}.\]

4. (ii.2) If $h\geq 2$ and $ph(h(1-p)/(h-1))^{h-1}>\mathrm{e}^{-1}$ , then the sequence $\{W_{\mathrm{binomial}}^{(\ell)}(C_\ell)\}_{\ell\geq 1}$ satisfies $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , where

   \[\Delta_\ell \;:\!=\;-\frac{\left(1+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm e^{\frac{1}{24\cdot25}}(1-p)}{p(h-1)\sqrt {2\pi}}\right)^{-1}\left[\log\left(ph\left( \frac{h(1-p)}{h-1}\right) ^{h-1}\right)\right]^3}{1.16}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}.\]

Proof. It is well known that the total progeny Z of a sub-critical Galton–Watson process with one ancestor and binomial offspring law with parameters (h, p) follows the Consul distribution, i.e.,

(45) \begin{equation}\mathbb P (Z=k)=\frac{1}{k}\binom{kh}{k-1}p^{k-1}(1-p)^{k(h-1)+1}, \quad k=1,2,\ldots.\end{equation}

By Stirling’s upper and lower bounds on the factorial, for $k\geq 1$ and $h\geq 2$ , we have

(46) \begin{align}&\binom{kh}{k-1}\nonumber\\[5pt] &=\frac{(kh)!}{(k-1)!(k(h-1)+1)!}\nonumber\\[5pt] &\leq\frac{1}{(k-1)!}\times\frac{\sqrt{2\pi kh}\left(\frac{kh}{\mathrm e}\right)^{kh}\mathrm{e}^{\frac{1}{12 kh}}}{\sqrt{2\pi (k(h-1)+1)}\left(\frac{k(h-1)+1}{\mathrm e}\right)^{k(h-1)+1}\mathrm{e}^{\frac{1}{12[k(h-1)+1]+1}}}\nonumber\\[5pt] &=\frac{\mathrm e^{-k+1}}{(k-1)!}\sqrt{\frac{kh}{k(h-1)+1}}\frac{(kh)^{kh}}{(k(h-1)+1)^{k(h-1)+1}}\mathrm{e}^{\frac{1}{12 kh}-\frac{1}{12[k(h-1)+1]+1}}\nonumber\\[5pt] &=\frac{\mathrm e^{-k+1}}{(k-1)!}\sqrt{\frac{kh}{k(h-1)+1}}\frac{(k(h-1))^{kh}}{(k(h-1)+1)^{k(h-1)+1}}\left(\frac{h}{h-1}\right)^{kh}\mathrm{e}^{\frac{1}{12 kh}-\frac{1}{12[k(h-1)+1]+1}}\nonumber\\[5pt] &\leq\frac{\mathrm e^{-k+1}}{(k-1)!}\sqrt{1+\frac{1}{h-1}}\frac{(k(h-1))^{kh}}{(k(h-1)+1)^{k(h-1)+1}}\left(\frac{h}{h-1}\right)^{kh}\mathrm{e}^{\frac{1}{12 kh}-\frac{1}{12[k(h-1)+1]+1}}\\[5pt] &=\frac{\mathrm e^{-k+1}}{(k-1)!}\sqrt{1+\frac{1}{h-1}}\left(\frac{k(h-1)}{k(h-1)+1}\right)^{k(h-1)+1}[k(h-1)]^{k-1}\left(\frac{h}{h-1}\right)^{kh}\mathrm{e}^{\frac{1}{12 kh}-\frac{1}{12[k(h-1)+1]+1}}\nonumber\\[5pt] &\leq\frac{\mathrm{e}^{\frac{1}{24\cdot 25}}}{h-1}\frac{\mathrm e^{-k+1} k^{k-1}}{(k-1)!}\sqrt{1+\frac{1}{h-1}}\mathrm e ^{-1}\left(\frac{h^h}{(h-1)^{h-1}}\right)^{k},\nonumber\end{align}

where the inequality (46) follows if we notice that $kh/[k(h-1)+1]\leq h/(h-1)$ , and the last inequality follows if we notice that

\[\mathrm{e}^{\frac{1}{12 kh}-\frac{1}{12[k(h-1)+1]+1}}\leq\mathrm{e}^{\frac{1}{24\cdot 25}}, \quad\text{for each $k\geq 1$ and $h\geq 2$,}\]

and that $\left(1\pm\frac{1}{n}\right)^n \leq\mathrm e^{\pm 1},$ $n\geq 1$ . By this latter inequality (with the sign $+$ ) and the Stirling lower bound on the factorial, we have

\[\frac{\mathrm{e}^{-k+1}}{(k-1)!} k^{k-1} \leq \frac{\mathrm e}{\sqrt {2\pi (k-1)}},\quad k\geq 2.\]

Therefore, for $k\geq 2$ and $h\geq 2$ , we have

(47) \begin{equation}\binom{kh}{k-1}\leq\frac{\mathrm{e}^{\frac{1}{24\cdot 25}}}{h-1}\frac{1}{\sqrt{2\pi(k-1)}}\sqrt{1+\frac{1}{h-1}}\left(\frac{h^h}{(h-1)^{h-1}}\right)^{k}\;=\!:\;C_{h,k}.\end{equation}

We now distinguish between two cases: $h=1$ and $h\geq 2$ .

Case $\textit{h=1.}$

Since $\binom{kh}{k-1}=k$ , by (45), for any $m\in\mathbb N$ we have

\[\mathbb E Z^m=\frac{1-p}{p}\sum_{k\geq 1} k^m p^{k}=\frac{1-p}{p}\sum_{k\geq 1} k^m\mathrm{e}^{-\nu_1 k},\quad\text{where $\nu_1\;:\!=\;-\log p>0$.}\]

Using Lemma 3 we have

(48) \begin{equation}\frac{{\mathbb E} Z^m}{\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}} \leq \frac{(1-p)m!}{p\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}}\left(\frac{1}{\nu_1^{m+1}}+\frac{1}{\pi m m!} + \frac{2}{\pi^{m+1}} \right),\quad m,\ell\in\mathbb N.\end{equation}

Proof of Part (i.1).

If $p\leq\mathrm{e}^{-1}$ , then $\nu_1 \geq 1$ ; therefore, setting $u_p\;:\!=\;1.05 \frac{1-p}{p}$ , by (48), for any $m\geq 3$ and $\ell\in\mathbb N$ , we have

(49) \begin{align}\frac{{\mathbb E} Z^m}{\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}} &\leq \frac{m!}{\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}}1.05 \frac{1-p}{p}\nonumber\\[5pt] &=\frac{m!}{\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}}u_p\nonumber\\[5pt] &=\frac{m!}{((u_p^{-1})\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}}(u_p^{-1})^{m-3}\nonumber\\[5pt] &\leq\frac{m!}{((u_p^{-1})\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}},\end{align}

where the latter inequality follows if we notice that $u_p^{-1}\leq 1$ (indeed $\frac{1}{p}-1 \geq \log \left( \frac{1}{p}\right) \geq 1$ ), and so $u_p\geq 1$ . Combining (49) with (34), we have that the condition (22) is satisfied with $\Delta_\ell \;:\!=\; \frac{p}{1.05(1-p)}\sqrt{\lambda_\ell \text{Leb}(B_\ell\cap C_\ell)}$ , and the claim follows by Corollary 2.

Proof of Part (i.2).

If $p>\mathrm{e}^{-1}$ , then $\nu_1<1$ ; therefore, setting $u_p\;:\!=\;\frac{1-p}{p\nu_1^3}$ , by (48), for any $m\geq 3$ and $\ell\in\mathbb N$ ,

\begin{align*} \frac{{\mathbb E} Z^m}{\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}} &\leq \frac{(1-p)m!}{p(\nu_1\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}}\nu_1^{m-2}\left(\frac{1}{\nu_1^{m+1}}+\frac{1}{\pi m m!} + \frac{2}{\pi^{m+1}} \right)\\[5pt] &= \frac{1-p}{p\nu_1^3} \frac{m!}{(\nu_1\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}}\left(1+\frac{\nu_1^{m+1}}{\pi m m!} + 2\left(\frac{\nu_1}{\pi}\right)^{m+1} \right)\\[5pt] &\leq \frac{1-p}{p\nu_1^3} \frac{m!}{(\nu_1\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}}1.05\\[5pt] &=u_p\frac{m!}{(\nu_1\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}}1.05\\[5pt] &=[(1.05 u_p)^{-1}]^{m-3}\frac{m!}{((1.05 u_p)^{-1}\nu_1\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}}.\end{align*}

Since the function $(\mathrm{e}^{-1},1)\ni p\mapsto u_p=\frac{1-p}{-p(\log p)^3}$ is increasing and $\lim_{p\to\mathrm{e}^{-1}}u_p=\mathrm{e}-1>1$ , we have that $u_p>1$ and so $1.05 u_p > 1$ , $p\in (\mathrm{e}^{-1},1)$ . Therefore, for all $m\geq 3$ and $\ell\in\mathbb N$ , we have

\begin{align*} \frac{{\mathbb E} Z^m}{\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)}^{m-2}} &\leq \frac{m!}{((1.05 u_p)^{-1}\nu_1\sqrt{\lambda_\ell \mathrm{Leb}(B_\ell \cap C_\ell)})^{m-2}}.\end{align*}

Combining this latter inequality with (34), we have that the condition (22) is satisfied with $\Delta_\ell \;:\!=\; \frac{p(\log p)^4}{1.05(1-p)}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}$ , and the claim follows by Corollary 2.

$\textit{Case}\,\,{h\geq 2.}$

If $h\geq 2$ , then by (47) we have

\[\binom{kh}{k-1}\leq\mathrm{11}_{\{k=1\}}+C_{h,k}\mathrm{11}_{\{k\geq 2\}}.\]

Combining this relation with (45) we have

\begin{align*} {\mathbb E} Z^m &\leq (1-p)^h+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm{e}^{\frac{1}{24\cdot 25}}}{(h-1)\sqrt {2\pi}}\sum_{k\geq 2} k^{m-1} \frac{1}{\sqrt {k-1}}\!\left(\!\frac{h^h}{(h-1)^{h-1}} \!\right)^k\! p^{k-1} (1-p)^{k(h-1)+1}\\[5pt] &=(1-p)^h+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm{e}^{\frac{1}{24\cdot 25}}(1-p)}{p(h-1)\sqrt {2\pi}}\sum_{k\geq 2} k^{m-1} \frac{1}{\sqrt {k-1}}\left(\frac{h^h}{(h-1)^{h-1}} \right)^kp^{k} (1-p)^{k(h-1)}\\[5pt] &=(1-p)^h+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm{e}^{\frac{1}{24\cdot 25}}(1-p)}{p(h-1)\sqrt {2\pi}}\sum_{k\geq 2} k^{m-1} \frac{1}{\sqrt {k-1}}\left(ph \left(\frac{h(1-p)}{h-1}\right) ^{h-1} \right)^k\nonumber\\[5pt] &\leq(1-p)^h+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm{e}^{\frac{1}{24\cdot 25}}(1-p)}{p(h-1)\sqrt {2\pi}}\sum_{k\geq 2} k^{m-1}\mathrm{e}^{-\nu_2 k},\quad\text{$m\geq 3$},\nonumber\end{align*}

where

\[\nu_2\;:\!=\;-\log\left(ph\left( \frac{h(1-p)}{h-1}\right) ^{h-1}\right).\]

Now we are going to verify that $\nu_2>0$ , i.e.,

(50) \begin{equation}ph\left( \frac{h(1-p)}{h-1}\right) ^{h-1}<1.\end{equation}

Setting $x\;:\!=\;ph \in (0,1)$ , we have

\begin{align*} x\left( \frac{h-x}{h-1}\right)^{h-1}&=x\left( \frac{h-1+1-x}{h-1}\right)^{h-1}\\[5pt] &=x\left( 1+\frac{1-x}{h-1}\right)^{h-1}\\[5pt] &= x\mathrm e^{(h-1)\log \left(1+ \frac{1-x}{h-1}\right)}\\[5pt] &\leq x\mathrm e^{1-x}.\end{align*}

The relation (50) follows if we notice that the mapping $x\in (0,1)\mapsto x\mathrm e^{1-x}$ is an increasing bijection from (0,1) to itself. Therefore, by Lemma 3, for any $m\geq 3$ , we have

(51) \begin{align}{\mathbb E} Z^m&\leq(1-p)^h+(m-1)!\sqrt{1+\frac{1}{h-1}} \frac{\mathrm{e}^{\frac{1}{24\cdot 25}}(1-p)}{p(h-1)\sqrt {2\pi}}\left(\nu_2^{-m}+\frac{1}{\pi (m-1)(m-1)!}+\frac{2}{\pi^m}\right)\nonumber\\[5pt] &\leq(m-1)!\left((1-p)^h+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm{e}^{\frac{1}{24\cdot 25}}(1-p)}{p(h-1)\sqrt {2\pi}}\right)\left(\nu_2^{-m}+\frac{1}{\pi (m-1)(m-1)!}+\frac{2}{\pi^m}\right)\nonumber\\[5pt] &\leq(m-1)!u_{p,h}\left(\nu_2^{-m}+\frac{1}{\pi (m-1)(m-1)!}+\frac{2}{\pi^m}\right),\end{align}

where

\begin{equation*}u_{p,h}\;:\!=\;1+\sqrt{1+\frac{1}{h-1}} \frac{\mathrm{e}^{\frac{1}{24\cdot 25}}(1-p)}{p(h-1)\sqrt {2\pi}}.\end{equation*}

Proof of Part (ii.1).

If $ph(h(1-p)/(h-1))^{h-1}\leq\mathrm{e}^{-1}$ , i.e., $\nu_2\geq 1$ , then combining (36) (with $\nu_2$ in place of $\nu$ ) with (51), we have

(52) \begin{align}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell\cap C_\ell)}^{m-2} }&\leq\frac{((u_{p,h})^{-1})^{m-3}m!}{((u_{p,h})^{-1}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)})^{m-2}}\nonumber\\[5pt] \leq & \frac{m!}{((u_{p,h})^{-1}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)})^{m-2}},\quad m\geq 3, \end{align}

where we used that $(u_{p,h})^{-1}<1$ . Combining (52) with (34), we have that the condition (22) is satisfied with $\Delta_\ell\;:\!=\;(u_{p,h})^{-1}\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}$ , and the claim follows by Corollary 2.

Proof of Part (ii.2).

If $ph(h(1-p)/(h-1))^{h-1}>\mathrm{e}^{-1}$ , i.e., $\nu_2<1$ , then combining (37) (with $\nu_2$ in place of $\nu$ ) with (51), we have

\begin{align*}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}}& \leq u_{p,h}\frac{(m-1)!}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}}\left(\nu_2^{-m}+0.16\right)\nonumber\\[5pt] &=\frac{(m-1)!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu_2)^{m-2}}\frac{\nu_2^m}{(u_{p,h})^{-1}\nu_2^2}\left(\nu_2^{-m}+0.16\right),\quad m\geq 3.\nonumber\end{align*}

Since $\nu_2 <1$ , we have

\begin{align*} \frac{\nu_2^m}{(u_{p,h})^{-1}\nu_2^2}\left(\nu_2^{-m}+0.16\right)\nonumber&\leq\frac{1.16}{(u_{p,h})^{-1}\nu_2^2}\;:\!=\;\widetilde{u}_{p,h}.\end{align*}

Therefore

\begin{align*}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}}&\leq\frac{(m-1)!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu_2)^{m-2}}\widetilde{u}_{p,h}\nonumber\\[5pt] &=\frac{(m-1)!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu_2 (\widetilde{u}_{p,h})^{-1})^{m-2}}((\widetilde{u}_{p,h})^{-1})^{m-3},\quad m\geq 3.\end{align*}

Using Lemma 4, we have that $(\widetilde{u}_{p,h})^{-1} <(u_{p,h})^{-1}\nu_2^2<(u_{p,h})^{-1}<1$ . Therefore

\begin{align*}\frac{\mathbb E Z^m}{\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}^{m-2}} \leq & \frac{m!}{(\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu_2(\widetilde{u}_{p,h})^{-1})^{m-2}},\quad m\geq 3.\end{align*}

Combining this latter inequality with (34), we have that the condition (22) is satisfied with $\Delta_\ell \;:\!=\;\sqrt{\lambda_\ell \text{Leb}(B_\ell \cap C_\ell)}\nu_2(\widetilde{u}_{p,h})^{-1}$ , and the claim follows by Corollary 2.

8.1. Gaussian approximation

In this section we apply Theorem 3 to the interference $I(\{\textbf{0}\})$ (see e.g. Remark 1) when the Poisson process of node locations has a piecewise constant intensity function of the form $\lambda(x)\;:\!=\;\lambda\textbf{1}_B(x)$ , for some $\lambda>0$ and $B\in\mathcal{B}(\mathbb R^2)$ . In such a case we have quite explicit upper bounds on the Wasserstein and Kolmogorov distances. The following corollary (whose proof is straightforward, and therefore omitted) allows for explicit bounds for some classes of signal power distributions and attenuation functions.

Corollary 7. Let $\lambda(x)\;:\!=\;\lambda\textbf{1}_B(x)$ , $x\in\mathbb R^2$ , for some $\lambda>0$ and $B\in \mathcal{B}(\mathbb R^2)$ such that

\[0<\mathbb E Z_1^2\int_B A^2(x)\mathrm{d}x<\infty.\]

Then

\[d_W(L\{\textbf{0}\}),G)\leq\frac{1}{\sqrt\lambda}\frac{\mathbb E Z_1^3}{(\mathbb E Z_1^2)^{3/2}}\frac{\int_B A(x)^3\mathrm{d}x}{\left(\int_B A(x)^2\mathrm{d}x\right)^{3/2}}\]

and

\begin{align}&d_K(L(\{\textbf{0}\}),G)\nonumber\\[5pt] &\quad\leq\frac{1}{\sqrt\lambda}\left[1+\frac{1}{2}\max\Biggl\{4,\left[4\frac{1}{\lambda}\frac{\mathbb E Z_1^4}{(\mathbb E Z_1^2)^2}\frac{\int_{B}A(x)^4\mathrm{d}x}{\left(\int_{B}A(x)^2\mathrm{d}x\right)^2}+2\right]^{1/4}\Bigg\}\right]\frac{\mathbb E Z_1^3}{(\mathbb E Z_1^2)^{3/2}}\frac{\int_{B}A(x)^3\mathrm{d}x}{\left(\int_{B}A(x)^2\mathrm{d}x\right)^{3/2}}\nonumber\\[5pt] &\qquad\qquad\qquad+\frac{1}{\sqrt\lambda}\frac{\sqrt{\mathbb E Z_1^4}}{\mathbb E Z_1^2}\frac{\sqrt{\int_{B}A(x)^4\mathrm{d}x}}{\int_{B}A(x)^2\mathrm{d}x}.\nonumber\end{align}

Example 2. If the path loss function is the Hertzian attenuation function, i.e., $A(x)\;:\!=\;\max\{R,\|x\|\}^{-\alpha}$ , $x\in{\mathbb R}^2$ , for some $R>0$ and $\alpha>1$ , $B\;:\!=\;{\mathbb R}^2$ , and ${\mathbb E} Z_1^2\in (0,\infty)$ , then Corollary 7 applies with

\[\int_B A(x)^m\mathrm{d}x=2\pi\left(\frac{1}{2}-\frac{1}{2-\alpha m}\right)R^{2-\alpha m},\quad\text{for $m=2,3,4$.}\]

8.2. Moderate deviations, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction term

In this section we apply Theorem 4 to the sequence $\{I_\ell(\{\textbf{0}\})\}_{\ell\geq 1}$ (defined in Remark 2) when the Poisson processes of node locations have piecewise deterministic intensity functions of the form $\lambda_\ell(x)\;:\!=\;\lambda_\ell\textbf{1}_{B_\ell}(x)$ , $x\in\mathbb R^2$ , for some sequences $\{\lambda_\ell\}_{\ell\geq 1}\subset (0,\infty)$ and $\{B_\ell\}_{\ell\geq 1}\subset\mathcal{B}(\mathbb R^2)$ . In such a case the assumption (17) is greatly simplified. The following corollary (whose proof is straightforward, and therefore omitted) holds.

Corollary 8. Let $\{B_\ell\}_{\ell \in \mathbb N}\subset\mathcal{B}(\mathbb R^d)$ and $\{\mathbb Q_\ell\}_{\ell\geq 1}$ be such that

(53) \begin{equation}0<\mathbb E (Z_1^{(\ell)})^2\int_{B_\ell}A(x)^2\mathrm{d}x<\infty,\quad \ell\geq 1,\end{equation}

and assume that there exist a non-negative constant $\gamma\geq 0$ and a positive numerical sequence $\{\Delta_\ell\}_{\ell \in \mathbb N}$ such that

(54) \begin{equation}\frac{1}{\lambda_\ell^{\frac{m}{2}-1}}\frac{\mathbb E(Z_1^{(\ell)})^m}{(\mathbb{E}(Z_1^{(\ell)})^2)^{m/2}}\frac{\int_{B_\ell}A(x)^m\mathrm{d}x}{\left(\int_{B_\ell}A(x)^2\mathrm{d}x\right)^{m/2}}\leq \frac{(m!)^{1+\gamma}}{\Delta_\ell ^{m-2}},\quad \text{for all } m\geq 3 \text{ and }\ell\in\mathbb N.\end{equation}

Then the sequence $\{I_\ell(\{\textbf{0}\})\}_{\ell\geq 1}$ satisfies an $\textbf{MDP}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , a $\textbf{BCI}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , and an $\textbf{NACC}(\gamma,\{\Delta_\ell\}_{\ell\in\mathbb N})$ .

Example 3. Under the notation of Corollary 8, let us set $B_\ell\equiv\mathbb R^2$ for any $\ell\geq 1$ , suppose that $\mathbb Q_\ell$ is the exponential law with mean $\mu^{-1}$ , for some $\mu>0$ , and assume that the attenuation of the signal is Hertzian, i.e., $A(x)\;:\!=\;\max\{R,\|x\|\}^{-\alpha}$ , $x\in\mathbb R^2$ , for some constants $R>0$ and $\alpha>1$ . Then

\[\frac{\mathbb E(Z_1^{(\ell)})^m}{(\mathbb{E}(Z_1^{(\ell)})^2)^{m/2}}=\frac{m!}{2^{m/2}}\quad\text{and}\quad\int_{{\mathbb R}^2}A(x)^m\mathrm{d}x=\frac{\pi\alpha m}{\alpha m-2}R^{2-\alpha m},\quad\text{for any $\ell\in\mathbb N$ and $m\geq 2$.}\]

So the assumption (53) of Corollary 8 is satisfied, and the left-hand side of the relation (54) reads, for any $\ell\in\mathbb N$ and $m\geq 3$ ,

\[\frac{m!}{\left(R\sqrt{2\lambda_\ell\frac{\pi\alpha}{\alpha-1}}\right)^{m-2}}\times\frac{m(2\alpha-2)}{4(\alpha m-2)}.\]

Since $m\geq 3$ we have $\frac{\alpha m-m}{\alpha m-2}<1.$ So Corollary 8 yields that the sequence $\{I_\ell(\{\textbf{0}\})\}_{\ell\geq 1}$ satisfies an $\textbf{MDP}(0,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , a $\textbf{BCI}(0,\{\Delta_\ell\}_{\ell\in\mathbb N})$ , and an $\textbf{NACC}(0,\{\Delta_\ell\}_{\ell\in\mathbb N})$ with

\[\Delta_\ell\;:\!=\;R\sqrt{2\lambda_\ell\frac{\pi\alpha}{\alpha-1}},\quad\ell\geq 1.\]