1.1. Introduction

Poisson cluster processes are an important class of point process models (see [Reference Bordenave and Torrisi5]). They occur frequently in applications such as cellular networks [Reference Chun, Hasna and Ghrayeb6], [Reference Tabassum, Hossain and Hossain23], insurance [Reference Jessen, Mikosch and Samorodnitsky18], [Reference Matsui19], [Reference Matsui and Mikosch20], queueing theory [Reference Fasen11], [Reference Fasen and Samorodnitsky12], [Reference Faÿ, González-Arévalo, Mikosch and Samorodnitsky13], and cosmology [Reference Neyman and Scott22]. Theoretical studies of Poisson cluster processes have attracted considerable attention; for example, see [Reference Bogachev and Daletskii4] for quasi-invariance, [Reference Faÿ, González-Arévalo, Mikosch and Samorodnitsky13] for some asymptotic behaviors of a Poisson cluster process, and [Reference Bordenave and Torrisi5] for functional large deviation principles for a large class of the processes. Linear Hawkes processes [Reference Hawkes15] are a class of Poisson cluster processes. They are amenable to statistical analysis. Thus they have a wide range of applications in large networks [Reference Delattre, Fournier and Hoffmann8], finance [Reference Hawkes16], and many other fields. The asymptotic behaviors of Hawkes processes have been widely studied; for example, see [Reference Bacry, Delattre, Hoffmann and Muzy1] for a functional central limit theorem and [Reference Bordenave and Torrisi5] for functional large deviation principles.

This paper considers functional central limit theorems and moderate deviations for Poisson cluster processes. Maximal inequalities, such as Ottaviani’s inequality and the maximal inequalities for martingales [Reference Billingsley3], play a very important role in functional limits. In this paper, we first propose a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. As an application, we obtain a functional moderate deviation principle for linear Hawkes processes.

The paper is organized as follows. In the remainder of this section, we give some preliminaries on Poisson cluster processes, present a maximal inequality for Poisson cluster processes, and state the main results of this paper. The proofs of the functional central limit theorem and the functional moderate deviation principle are given in Sections 2 and 3, respectively. The proof of the maximal inequality is postponed to the appendix.

1.2. Main results

Let us recall some basic definitions related to Poisson cluster processes (see [Reference Bordenave and Torrisi5]). A Poisson cluster process $\mathbf{X}\subset \mathbb{R}$ is a point process generated from an immigrant process and a family of offspring processes. The formal definition of a Poisson cluster process is the following:

1. (i) The immigrant process I is a homogeneous Poisson process with points $X_i\in \mathbb{R}$ and intensity constant $\nu\in (0,\infty)$.

2. (ii) Each immigrant $X_i$ generates a cluster, i.e. an offspring process $C_i=C_{X_i}$ which is a finite point process.

3. (iii) Given the immigrants, the centered clusters

   \begin{equation*}C_i-X_i=\{Y-X_i\,{:}\,Y\in C_i\}, \qquad X_i\in I,\end{equation*}
   are independent and identically distributed (i.i.d.) and independent of I.

4. (iv) $\mathbf{X}$ consists of the union of all the clusters.

For a point process $\mathbf{Y}$ on $\mathbb{R}$, let $N_{\mathbf{Y}}(0,t]$ denote the number of points of $\mathbf{Y}$ in the interval (0, t]. A process $\mathbf{Y}$ is called stationary if its law is translation-invariant; it is said to be ergodic if it is stationary with a finite intensity $\mathbb{E}(N_{\mathbf{Y}}(0,1])$, and

\begin{equation*}\lim_{t\rightarrow \infty}\frac{N_{\mathbf{Y}}(0,t]}{t}=\mathbb{E}(N_{\mathbf{Y}}(0,1])\end{equation*}

almost surely.

Let S denote the number of points in a cluster and assume $\mathbb{E}(S)<\infty$. Let $C_0$ be the cluster generated by an immigrant at 0, and let $L=\sup_{Y\in C_0}|Y|$ be the radius of $C_0$. Then we can see from the definition of a Poisson cluster process that $\mathbf{X}$ is ergodic with finite intensity $\nu \mathbb{E}(S)$.

First, let us present a maximal inequality which will play an important role in studying functional limits of Poisson cluster processes.

Lemma 1. Define

\begin{equation*}C(t)=\sum_{X_k\in I_{|(0,t]}}N_{C_k}(0, t].\end{equation*}

Let $0\leq s< t$ and $s=t_0<t_1<\cdots<t_n=t$. Then for any $r>0$,

(1)\begin{align}& \mathbb{P}\Big( \max_{1\leq l\leq n}\left| C(t_l)-C(s)-\mathbb{E}(C(t_l)-C(s)) \right|>3r \Big)\nonumber\\ &\quad \leq 2 \mathbb{P}\Bigg( \max_{0\leq l\leq n-1}\Bigg| \sum_{X_k\in I_{|(0,t_l]}}N_{C_k}(t_l, t_n]- \mathbb{E}\Bigg( \sum_{X_k\in I_{|(0,t_l]}}N_{C_k}(t_l, t_n]\Bigg)\Bigg|>r/2\Bigg)\\ &\qquad + 2\max_{0\leq l\leq n-1}\mathbb{P}\Bigg(\Bigg| \sum_{X_k\in I_{|(t_l,t_n]}}N_{C_k}(0, t_n]- \mathbb{E}\Bigg( \sum_{X_k\in I_{|(t_l,t_n]}}N_{C_k}(0, t_n]\Bigg)\Bigg|> r/2\Bigg).\nonumber\end{align}

The proof of the inequality is given in the appendix.

Now let us state our main results. Let D[0, 1] denote the space of càdlàg functions on the interval [0, 1] and let $AC_0[0,1]$ be the family of all absolutely continuous functions f with $f(0)=0$. Set $\|\,f\|=\sup_{t\in [0,1]}|\,f(t)|$. Let $\rho_s$ denote the Skorokhod topology on D[0, 1].

Theorem 1. Assume that

(A1)\begin{equation}\mathbb{E}((1+L)S^2)<\infty \end{equation}

holds. Then as $\alpha \to \infty$,

\begin{equation*}\frac{N_{\mathbf{X}}(0,\alpha t]-\mathbb{E}(N_{\mathbf{X}}(0,\alpha t])}{\sqrt{\alpha}} \stackrel{d} {\rightarrow} \sigma B(t) \mbox{ in } (D[0,1], \rho_s),\end{equation*}

where $\sigma ^2=\nu \mathbb{E}(S^2)$, $\{B(t),t\geq 0\}$ is the standard Brownian motion, and $\stackrel{d} {\rightarrow}$ denotes convergence in distribution.

Theorem 2. Let $\{b(\alpha),\alpha>0\}$ be a positive function satisfying

\begin{equation*}\lim_{\alpha\rightarrow \infty}\frac{b(\alpha)}{\alpha}= 0,\quad\lim_{\alpha\rightarrow \infty}\frac{b(\alpha)}{\sqrt {\alpha}}=+\infty. \eqno{(\text{SC})}\end{equation*}

Assume that the Cramér condition holds, i.e., that there exists a constant $\theta_0>0$ such that

(A2)\begin{equation}\mathbb{E}\left((1+L) e^{\theta_0 S}\right)<\infty.\ \end{equation}

Then

\begin{equation*}\left\{\frac{N_{\mathbf{X}}(0,\alpha t]-\mathbb{E}(N_{\mathbf{X}}(0,\alpha t])}{b(\alpha)},t\in[0,1]\right\}\end{equation*}

satisfies the large deviation principle (LDP) on $(D[0,1],\|\cdot\|)$ with speed $\frac{b^2(\alpha)}{\alpha}$ and good rate function $J\,{:}\,D[0,1]\to [0,\infty]$ defined by

(2)\begin{equation}J(\,f)=\left\{\begin{array}{ll}\displaystyle\frac{1}{2\sigma^2}\int_0^1{|\,\dot{f}(t)|^2}dt & \mbox{ if }f(t)=\int_0^t \dot{f}(s)ds \in AC_0[0,1];\\\hbox{}\\+\infty &\mbox{ otherwise}.\end{array}\right.\end{equation}

That is,

1. (1) for any $l\leq 0$, $\{f;\, J(\,f)\leq l\}$ is compact in $(D[0,1],\|\cdot\|)$;

2. (2) for any closed F in $(D[0,1],\|\cdot\|)$,

   \begin{equation*} \limsup_{\alpha\rightarrow \infty}\frac{\alpha}{b^2(\alpha)}\log \mathbb{P}\!\left(\frac{N_{\mathbf{X}}(0,\alpha\cdot]-\mathbb{E}(N_{\mathbf{X}}(0,\alpha\cdot])}{b(\alpha)}\in F\right)\leq -\inf_{f\in F}J(\,f), \end{equation*}
   and for any open G in $(D[0,1],\|\cdot\|)$,
   \begin{equation*} \liminf_{\alpha\rightarrow \infty}\frac{\alpha}{b^2(\alpha)}\log \mathbb{P}\!\left(\frac{N_{\mathbf{X}}(0,\alpha\cdot]-\mathbb{E}(N_{\mathbf{X}}(0,\alpha\cdot])}{b(\alpha)}\in G\right)\geq -\inf_{f\in G}J(\,f). \end{equation*}

Next, let us apply Theorems 1 and 2 to linear Hawkes processes. A Poisson cluster process $\mathbf{X}$ is called a Hawkes process if each cluster $C_i=C_{X_i}$ is a random set formed by the points of generations $n=0,1,\cdots$ with the following branching structure:

1. • The immigrant $X_i$ is said to be of generation 0.

2. • Given generations $0,1,\cdots,n$ in $C_i$, each point $Y\in C_i$ of generation n generates a Poisson process on $(Y,\infty)$ of offspring of generation $n+1$ with intensity function $h(\cdot-Y)$, where $h\,{:}\,(0,\infty)\rightarrow [0,\infty)$ is a nonnegative Borel function.

As usual, we assume that the mean number of points in any offspring process satisfies

(B1)\begin{equation}\mu\coloneqq \int_0^{\infty} h(t) dt\in (0,1). \end{equation}

We also assume that

(B2)\begin{equation}\int_0^{\infty} t h(t)dt<\infty. \end{equation}

It is known (cf. Bordenave and Torrisi [Reference Bordenave and Torrisi5]) that for a linear Hawkes process, the distribution of S is given by

\begin{equation*}\mathbb{P}(S=k)=\frac{e^{-k\mu}(k\mu)^{k-1}}{k!}, \quad k=1,2,\cdots.\end{equation*}

From the proof of Theorem 3.2.1 in [Reference Bordenave and Torrisi5], under the conditions (B1) and (B2), there exists a constant $0<\theta_0< \mu-1-\log \mu$ such that $\mathbb{E}\left((1+L) e^{\theta_0 S}\right)<\infty$; that is, the condition (A2) holds. Furthermore, the function

\begin{equation*}F(\theta)\coloneqq \mathbb{E}(e^{\theta S}), \qquad \theta < \mu-1-\log \mu,\end{equation*}

satisfies the equation $F(\theta)=e^{\theta}\exp\{\mu(F(\theta)-1)\}.$ Thus,

\begin{align*} F'(\theta) & =\frac{F(\theta)}{1-\mu F(\theta)},\\[4pt] \mathbb{E}(S) & =F'(0)=1/(1-\mu)<\infty,\end{align*}

and

\begin{equation*}\hspace*{-25pt}\mathbb{E}(S^2)=F''(0)=\frac{1}{(1-\mu)^3}.\end{equation*}

The following results are consequences of Theorems 1 and 2.

Corollary 1. Assume that (B1) and (B2) hold.

1. (1)

   \begin{equation*}\bigg\{\frac{N_{\mathbf{X}}(0,\alpha t]-\mathbb{E}(N_{\mathbf{X}}(0,\alpha t])}{\sqrt{\alpha}},t\in[0,1]\bigg\}\end{equation*}

converges in distribution to

\begin{equation*}\sqrt{\frac{\nu}{(1-\mu)^3}}B(t)\end{equation*}

in $(D[0,1],\rho_s)$.

1. (2)

   \begin{equation*}\bigg\{\frac{N_{\mathbf{X}}(0,\alpha t]-\mathbb{E}(N_{\mathbf{X}}(0,\alpha t])}{b(\alpha)},\ t\in[0,1]\bigg\}\end{equation*}

satisfies the LDP on $(D[0,1],\|\cdot\|)$with speed $\frac{b^2(\alpha)}{\alpha}$and good rate function $J^H(\,f)$defined by

\begin{equation*}J^H(\,f)=\left\{\begin{array}{l@{\quad}l}\displaystyle\frac{(1-\mu)^3}{2\nu }\int_0^1{|\,\dot{f}(t)|^2}dt & \mbox{ if } f\in AC_0[0,1],\\\hbox{}\\+\infty &\mbox{ otherwise}.\end{array}\right.\end{equation*}

Applying Corollary 1(2) to $F=\{f\in D[0,1]\,{;}\, f(0)=0,\ \|\,f\|\geq r\}$ for $r>0$ and $b(\alpha)=\alpha^{p}$ for $1/2<p<1$, we have

\begin{align*} &\lim_{\alpha\rightarrow \infty}\frac{1}{\alpha^{2p-1}}\log \mathbb{P}\left(\|N_{\mathbf{X}}(0,\alpha\cdot]-\mathbb{E}(N_{\mathbf{X}}(0,\alpha\cdot])\|\geq \alpha^{p}r\right)\\[4pt] &=-\inf_{f\in AC_0[0,1], \|\,f\|\geq r}\bigg\{\frac{(1-\mu)^3}{2\nu }\int_0^1{|\,\dot{f}(t)|^2}dt \bigg\}\\[4pt] &=-\frac{r^2(1-\mu)^3}{2\nu }.\end{align*}

For convenience, before we close this section, let us introduce some notation and state a Laplace integral formula for marked point processes (see Daley and Vere-Jones [Reference Daley and Vere-Jones7], Section 6.4).

1. • A marked point process N, with locations in a Polish space $\mathcal X$ and marks in a Polish space $\mathcal K$, is a point process $\{(X_i,\kappa_i)\}$ on $\mathcal X \times\mathcal K$ with the additional property that the process $ \{X_i\}$ is itself a point process. The process $\{X_i\}$ is referred to as the ground process.

2. • A marked point process N is said to have independent marks if, given the ground process $ \{X_i\}$, the $\{\kappa_i\}$ are mutually independent random variables such that the distribution of $\kappa_i$ depends only on the corresponding location $X_i$.

3. • Let N be a marked point process with i.i.d. marks and let $\{F(K|x)\,{:}\,K \in\mathcal B(\mathcal K), x \in\mathcal X\}$ be a kernel. This kernel is called the mark kernel of the process N if, for any $x\in\mathcal X$ and $K\in \mathcal K$, $F(K |x) =\mathbb{P}(\kappa_i\in K |X_i=x)$ almost surely.

4. • A marked point process N is called a marked Poisson process or a compound Poisson process if the process N has i.i.d. marks and the ground process is a Poisson process.

For the next lemma, see Proposition 6.4.IV and Lemma 6.4.VI in [Reference Daley and Vere-Jones7].

Lemma 2. A marked Poisson process that has mark kernel $F(\cdot |\cdot)$, and for which the Poisson ground process $N_g$ has intensity measure $\mu$, is equivalent to a Poisson process on the product space $\mathcal X \times\mathcal K$ with intensity measure $\Lambda(dx \times dz) = \mu(dx) F(dz | x)$. In particular, the probability generating functional of the process N is

\begin{equation*}G[h]=\mathbb{E}\!\left(\prod_{i}h(X_i,\kappa_i)\right)=\exp\left\{\int\int (h(x,\kappa)-1) F(dz | x)\mu(dx) \right\},\ h\in \mathcal V(\mathcal X \times\mathcal K),\end{equation*}

where $\mathcal V(\mathcal X \times\mathcal K)$ is the space of measurable functions $h(x,\kappa)$ such that $0\leq h(x,\kappa)\leq 1$ and, for some bounded set A, $h(x, \kappa)=1$ for all $\kappa\in K$ and $x\not\in A$.