Precise deviations for Cox processes with a shot noise intensity

TL;DR

This paper investigates the precise deviations of Cox processes with shot noise intensity, a model relevant in various fields, using the advanced mod-$\phi$ convergence method to derive new probabilistic results.

Contribution

It introduces a novel application of the mod-$\phi$ convergence method to analyze deviations in shot noise Cox processes, enhancing understanding of their probabilistic behavior.

Findings

Derived precise deviation results for shot noise Cox processes.

Applied mod-$\phi$ convergence to a new class of stochastic processes.

Provided tools for better risk assessment in fields like finance and insurance.

Abstract

We consider a Cox process with Poisson shot noise intensity which has been widely applied in insurance, finance, queue theory, statistic, and many other fields. Cox process is flexible because its intensity depends on not only the time but also a stochastic process, it can be considered as a two step randomization procedure. Due to the structure of such models, a number of useful and general results can easily be established. In this paper, we study precise deviations for shot noise Cox process using the recent mod-$\phi$ convergence method.

Equations

$${\lambda }_t=\nu +{\int }_0^{t}g(t-s)d{\bar{N}}_s,$$

$$e^{-t_n\eta (z)}\mathbb{E}[e^{z{X}_n}]\to \psi (z),$$

$$\displaystyle{\mathbb{P}}(X_{n}=t_{n}x)=\frac{e^{-t_{n}F(x)}}{\sqrt{2\pi t_{n}\eta^{\prime\prime}(\theta^{*})}}\Biggl{(}\psi(\theta^{*})+\frac{a_{1}}{t_{n}}+\frac{a_{2}}{t^{2}_{n}}+\cdots+\frac{a_{\nu-1}}{t^{\nu-1}_{n}}+\mathcal{O}\left(\frac{1}{t^{\nu}_{n}}\right)\Biggr{)},$$

$$\displaystyle{\mathbb{P}}(X_{n}\geq t_{n}x)=\frac{e^{-t_{n}F(x)}}{\sqrt{2\pi t_{n}\eta^{\prime\prime}(\theta^{*})}}\frac{1}{1-e^{-\theta^{*}}}\Biggl{(}\psi(\theta^{*})+\frac{b_{1}}{t_{n}}+\frac{b_{2}}{t_{n}}+\cdots+\frac{b_{\nu-1}}{t^{\nu-1}_{n}}+\mathcal{O}\left(\frac{1}{t^{\nu}_{n}}\right)\Biggr{)},$$

$$\mathbb{P}\left(X_n\ge t_n{\eta }^{\prime }(0)+\sqrt{t_n{\eta }^{\prime \prime }(0)}y\right)=\mathbb{P}\left({\mathcal{N}}_{\mathbb{R}}(0,1)\ge y\right)(1+o(1)).$$

$$\mathbb{P}\left(X_n\ge t_n{\eta }^{\prime }(0)+\sqrt{t_n{\eta }^{\prime \prime }(0)}y\right)=\frac{e^{-t_{n}F(x)}}{h\sqrt{2\pi t_n{\eta }^{\prime \prime }(h)}}(1+o(1))$$

$$\mathbb{P}\left(X_n\ge t_n{\eta }^{\prime }(0)+\sqrt{t_n{\eta }^{\prime \prime }(0)}y\right)=\frac{(1+o(1))}{y\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\exp \left(\frac{{\eta }^{\prime \prime \prime }(0)}{6({\eta }^{\prime \prime }(0){)}^{3/2}}\frac{y^3}{\sqrt{t_n}}\right)$$

$$\mathbb{P}\left(X_n\ge t_n{\eta }^{\prime }(0)+\sqrt{t_n{\eta }^{\prime \prime }(0)}y\right)=\frac{(1+o(1))}{y\sqrt{2\pi }}\exp \left(-\sum _{i=2}^{m-1}\frac{F^{(i)}({\eta }^{\prime }(0))}{i!}\frac{({\eta }^{\prime \prime }(0){)}^{i/2}{y}^i}{t_n^{(i-2)/2}}\right)$$

$$\displaystyle{\mathbb{P}}(N_{t}=tx)=e^{-tI(x)}\sqrt{\frac{I^{\prime\prime}(x)}{2\pi t}}\Biggl{(}\psi(\theta^{*})+\frac{a_{1}}{t}+\frac{a_{2}}{t^{2}}+\cdots+\frac{a_{\nu-1}}{t^{\nu-1}}+\mathcal{O}\left(\frac{1}{t^{\nu}}\right)\Biggr{)},$$

$$\psi (\theta ):=e^{\rho \phi (\theta )}$$

$$\displaystyle\varphi(\theta)=\int_{0}^{\infty}\Bigl{[}e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}-e^{(e^{\theta}-1)\int_{0}^{\infty}g(s)ds}\Bigr{]}du,$$

$$I(x)=\sup_{\theta\in{\mathbb{R}}}\{\theta x-\eta(\theta)\big{\}}$$

$$\eta(\theta)=(e^{\theta}-1)\nu+\rho\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}$$

$$\displaystyle{\mathbb{P}}(N_{t}\geq tx)=e^{-tI(x)}\sqrt{\frac{I^{\prime\prime}(x)}{2\pi t}}\frac{1}{1-e^{-\theta^{*}}}\Biggl{(}\psi(\theta^{*})+\frac{b_{1}}{t}+\frac{b_{2}}{t^{2}}+\cdots+\frac{b_{\nu-1}}{t^{\nu-1}}+\mathcal{O}\left(\frac{1}{t^{\nu}}\right)\Biggr{)},$$

$$\displaystyle e^{\eta(\theta)}=\exp\Bigl{[}(e^{\theta}-1)\nu+\rho\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\Bigr{]}={\mathbb{E}}[e^{\theta Y}],$$

$$\displaystyle\varphi(\theta)=\int_{0}^{\infty}\Bigl{[}{e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}}-e^{(e^{\theta}-1)\int_{0}^{\infty}g(s)ds}\Bigr{]}du,$$

$$e^{-t\eta (\theta )}\mathbb{E}[e^{\theta N_t}]\to \psi (\theta ):=e^{\rho \phi (\theta )},$$

$$\mathbb{P}\left(N_t\ge (\nu +\rho \Vert g{\Vert }_{L^1})t+\sqrt{t}\sqrt{\nu +\rho \Vert g{\Vert }_{L^1}(\Vert g{\Vert }_{L^1}+1)}y\right)=\Psi (y)(1+o(1)),$$

$$\mathbb{P}\left(N_t\ge (\nu +\rho \Vert g{\Vert }_{L^1})t+\sqrt{t}\sqrt{\nu +\rho \Vert g{\Vert }_{L^1}(\Vert g{\Vert }_{L^1}+1)}y\right)=\frac{e^{-tI(x^{\ast })}}{{\theta }^{\ast }\sqrt{2\pi t{\eta }^{\prime \prime }({\theta }^{\ast })}}(1+o(1)),$$

$$\begin{array}{r}\mathbb{P}\left(N_t\ge (\nu +\rho \Vert g{\Vert }_{L^1})t+\sqrt{t}\sqrt{\nu +\rho \Vert g{\Vert }_{L^1}(\Vert g{\Vert }_{L^1}+1)}y\right)\\ =\frac{(1+o(1))}{y\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}\exp \left(\frac{{\eta }^{\prime \prime \prime }(0)}{6({\eta }^{\prime \prime }(0){)}^{3/2}}\frac{y^3}{\sqrt{t}}\right)\end{array}$$

$$\begin{array}{r}\mathbb{P}\left(N_t\ge (\nu +\rho \Vert g{\Vert }_{L^1})t+\sqrt{t}\sqrt{\nu +\rho \Vert g{\Vert }_{L^1}(\Vert g{\Vert }_{L^1}+1)}y\right)\\ =\frac{(1+o(1))}{y\sqrt{2\pi }}\exp \left(-\sum _{i=2}^{m-1}\frac{I^{(i)}({\eta }^{\prime }(0))}{i!}\frac{({\eta }^{\prime \prime }(0){)}^{i/2}{y}^i}{t^{(i-2)/2}}\right),\end{array}$$

$$\displaystyle{\mathbb{E}}\left[e^{\theta N_{t}}\right]=\exp{\Biggl{[}(e^{\theta}-1)t\nu+\rho\int_{0}^{t}\Bigl{(}e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}-1\Bigr{)}du\Biggr{]}}$$

$$\begin{split}{\mathbb{E}}\left[e^{\theta N_{t}}\right]&={\mathbb{E}}\left[e^{(e^{\theta}-1)\int_{0}^{t}\lambda_{s}ds}\right]\\ &={\mathbb{E}}\left[e^{(e^{\theta}-1)\left[\nu t+\int_{0}^{t}\int_{0}^{s}g(s-u)d\bar{N}_{u}ds\right]}\right]\\ &={\mathbb{E}}\left[e^{(e^{\theta}-1)\left[\nu t+\int_{0}^{t}[\int_{u}^{t}g(s-u)ds]d\bar{N}_{u}\right]}\right]\\ &=e^{(e^{\theta}-1)t\nu}{\mathbb{E}}\left[e^{(e^{\theta}-1)\int_{0}^{t}[\int_{u}^{t}g(s-u)ds]d\bar{N}_{u}}\right]\\ &=e^{(e^{\theta}-1)t\nu}\exp\Biggl{[}\rho\int_{0}^{t}\left(e^{(e^{\theta}-1)\int_{u}^{t}g(s-u)ds}-1\right)du\Biggr{]}\\ &=e^{(e^{\theta}-1)t\nu}\exp\Biggl{[}\rho\int_{0}^{t}\left(e^{(e^{\theta}-1)\int_{0}^{t-u}g(s)ds}-1\right)du\Biggr{]}\\ &=\exp{\Biggl{[}(e^{\theta}-1)t\nu+\rho\int_{0}^{t}\Bigl{(}e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}-1\Bigr{)}du\Biggr{]}}.\end{split}$$

$$\displaystyle\lim_{t\rightarrow\infty}\frac{1}{t}\log{\mathbb{E}}\left[e^{\theta N_{t}}\right]=(e^{\theta}-1)\nu+\rho\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}.$$

$$\displaystyle I(x)=\sup_{\theta\in{\mathbb{R}}}\Biggl{\{}\theta x-\Bigl{[}(e^{\theta}-1)\nu+\rho\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\Bigr{]}\Biggr{\}}$$

$$\displaystyle\exp\Bigl{[}(e^{\theta}-1)\nu+\rho\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\Bigr{]}={\mathbb{E}}\left[e^{\theta Y}\right]$$

$$\begin{split}\mathbb{E}\left[e^{\theta Y}\right]&=\mathbb{E}\left[e^{\theta X}\right]\mathbb{E}\left[e^{\theta Z}\right]\\ &=\exp\left[(e^{\theta}-1)\nu\right]\exp\left[\rho\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\right]\\ &=\exp\Bigl{[}(e^{\theta}-1)\nu+\rho\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\Bigr{]}\end{split}$$

$$\displaystyle e^{-t\eta(\theta)}{\mathbb{E}}[e^{\theta N_{t}}]=\frac{{\mathbb{E}}\left[e^{\theta N_{t}}\right]}{\exp\Bigl{[}(e^{\theta}-1)\nu t+\rho t\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\Bigr{]}}.$$

$$\begin{split}&\frac{{\mathbb{E}}\left[e^{\theta N_{t}}\right]}{\exp\Bigl{[}(e^{\theta}-1)\nu t+\rho t\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\Bigr{]}}\\ &=\exp\Biggl{[}\rho\int_{0}^{t}\Bigl{(}e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}-1\Bigr{)}du-\rho t\Bigl{(}e^{(e^{\theta}-1)\|g\|_{L^{1}}}-1\Bigr{)}\Biggr{]}\\ &=\exp\Biggl{[}\rho\int_{0}^{t}\Bigl{[}e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}-e^{(e^{\theta}-1)\int_{0}^{\infty}g(s)ds}\Bigr{]}du\Biggr{]}\\ &\rightarrow\psi(\theta):=e^{\rho\varphi(\theta)}.\end{split}$$

$$\displaystyle\int_{0}^{t}\Bigl{[}e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}-e^{(e^{\theta}-1)\int_{0}^{\infty}g(s)ds}\Bigr{]}du$$

Full text

*MSC2010: * primary 60G55; secondary 60F05, 60F10.

Keywords: Cox processes, Poisson shot noise, Mod$-\phi$ convergence, Precise deviations.

1 Introduction

A Cox process is first introduced by Cox (1955) and a natural generalization of a Poisson process by considering the intensity of Poisson process as a realization of a random measure (M$\o$ller, 2003). The Cox process provides the flexibility of letting the intensity not only depend on time but also allowing it to be a stochastic process. Hence, it can be viewed as a two-step randomization procedure which can deal with the stochastic nature of catastrophic loss occurrences in the real world. Moreover, shot noise processes  (Cox, 1980) are particularly useful to model claim arrivals; they provide measures for frequency, magnitude and the time period needed to determine the effect of catastrophic events within the same framework; as time passes, the shot noise process decreases as more and more losses are settled, and this decrease continues until another event occurs which will result in a positive jump. Therefore, the shot noise process can be used as the intensity of a Cox process to measure the number of catastrophic losses. The shot noise Cox processes were introduced in M$\o$ller (2003) and further generalized in Hellmund, Prokešová, and Jensen (2008) without discussing the statistical inference for the model. Note that the class of shot noise Cox processes also includes the very popular Poisson Neyman-Scott processes like the Thomas process((Thomas, 1949),[(Illian et al., 2008), Section 6.3.2]).

The Cox model has been used widely in many aspects: insurance, finance, queue theory, statistic etc.(cf.(Albrecher and Asmussen, 2006),(Cox, 1955),(Dassios and Jang, 2003)). Dassios and Jang (2003) applied the Cox process with Poisson shot noise intensity to pricing stop-loss catastrophe reinsurance contract and catastrophe insurance derivatives. Albrecher and Asmussen (2006) studied the asymptotic estimates for infinite time and finite time ruin probabilities of the risk model.

There are many applications of large deviations such as insurance, portfolio management, risk management, queue system, statistics((Macci and Stabile, 2006), (Gao and Yan, 2008),(Shen, Lin, and Zhang, 2009),(Ganesh, Macci, and Torrisi, 2007)). Macci and Stabile (2006) studied the large deviation principles for the Markov modulated risk process with reinsurance. Gao and Yan (2008) extended the result considered in Macci and Stabile (2006) to the sample path large and moderate deviation principles. Shen, Lin, and Zhang (2009) obtained the precise large deviation results for the customer arrival based risk model which can be treated as a generalized Poisson shot noise process. Recently, Macci and Torrisi (2011) took into account the large deviation estimations for the ruin probability of the risk processes with shot noise Cox claim number process and reserve dependent premium rate. Many researchers have made great efforts of the precise large deviations for the loss process of a classical insurance risk model and have obtained a lot of inspiring results. Precise large deviations for the loss process have been widely investigated. For the classic results, we refer the reader to Cline and Hsing (1991), Klüppelberg and Mikosch (1997), Mikosch and Nagaev (1998), and Ng et al. (2004), among others. It is worth mentioning that the results of precise large deviations for random sums are particulary useful for evaluation of some risk measures such as conditional tail expectation and value at risk of aggregate claims of a large insurance portfolio; see McNeil et al. (2005) for a review of risk measures. Wang and Wang (2007) investigated precise large deviations of multi-risk models.

Previous works on insurance applications using a shot noise process or a Cox process with shot noise intensity can be found in Brémaud (2000), Dassios and Jang (2003), Jang and Krvavych (2004), Torrisi (2004), Albrecher and Asmussen (2006), Macci and Torrisi (2011), Zhu (2013) and Schmidt (2014). However, previous study on the large deviations and moderate deviations for Cox process only gives the leading order term (Gao and Yan, 2012), but not the higher order expansion which make more accurate computational tractability for an financial application. Distributional properties for such a model generate computational tractability for an financial application. Transform, distribution, and moment formulae computational tractability for a range of applications in portfolio credit risk. The transform formulae facilitate the valuation, hedging, and calibration of a portfolio credit derivative, which is a security whose payoff is a specified function of the portfolio loss, and which provides insurance against default losses in the portfolio.

In this paper, we assume that $\{N_{t},t\geq 0\}$ is a Cox process with Poisson shot noise intensity, that is, the intensity $\lambda_{t}$ of $N_{t}$ is stochastic and at time t, given by

[TABLE]

where $\bar{N}_{t}$ is a Poisson process with intensity $\rho,$ $\nu>0$ and $g:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}_{+}$ being locally bounded. We study precise deviations for shot noise Cox process using the recent mod-$\phi$ convergence method developed in Feray, Meliot, and Nikeghbali (2013). In many applications in finance, insurance, and other fields, more precise deviations are desired, which motivates us to study the precise deviations for Cox process with a shot noise due to higher order expansion given by the precise deviation for Cox process. Taylor expansions have numerous applications to simplify complex functions in finance such as the price of a portfolio of options or bonds. The more terms used in the expansion the more accurate the approximation. Higher order Taylor expansions are obtained using higher partial derivatives. Higher derivatives are derivatives of derivatives. Thus higher order expansion is one of the most important concepts in mathematical finance literature. In particular, using the re-normalization theory called mod-$\phi$ convergence method, we construct an extremely flexible framework to study the precise deviations and limit theorems. This type of convergence is a relatively new concept with many deep ramifications. It is worth mentioning that the results of precise large deviations for Cox process with a shot noise are particulary useful for calculations and evaluations in financial applications using the higher order expansion given by the precise deviations which is more applicable than the large or moderate deviations principle. Very recently, precise deviations for Hawkes processes was studied in Gao and Zhu (2017) using the mod-$\phi$ convergence method. In particular, Gao and Zhu (2017) used Hawkes processes for large time asymptotics, that strictly extends and improves the existing results in the literature. In order to apply the results of mod-$\phi$ convergence theory, we manipulate the the moment generating function for shot noise Cox process, and then the precise deviations principle and precise moderate deviation and fluctuation results are obtained by the mod-$\phi$ convergence method after careful analysis and series of lemmas.

The structure of this paper is organized as follows. Some auxiliary results and the main results are stated in Section 2. The proofs for the main theorems are contained in Section 3.

2 Statement of the main results

This section states the main results of this paper. It consists of two key lemmas and the precise deviations principle and furthermore precise deviation principle and fluctuation results. The main strategy of proving the precise deviations principle is by showing the mod-$\phi$ convergence as defined in Feray, Meliot, and Nikeghbali (2013) and apply their Theorem 3.4 and 3.9 to get the main Theorem. We start with the definition of mod-$\phi$ convergence and then the assumptions which we will use throughout the paper.

Let us first recall the definition of mod-$\phi$ convergence, see e.g. Definition 1.1. (Feray, Meliot, and Nikeghbali, 2013). Let $(X_{n})_{n\in{\mathbb{N}}}$ be a sequence of real-valued random variables and ${\mathbb{E}}[e^{zX_{n}}]$ exist in a strip $\mathcal{S}_{(c,d)}:=\{z\in\mathbb{C}:c<\mathcal{R}(z)<d\},$ with $c<d$ extended real numbers, i.e. we allow $c=-\infty$ and $d=+\infty$ and $\mathcal{R}(z)$ denotes the real part of $z\in\mathbb{C}$ throughout this paper. We assume that there exists a non-constant infinitely divisible distribution $\phi$ with $\int_{{\mathbb{R}}}e^{zx}\phi(dx)=e^{\eta(z)},$ which is well defined on $\mathcal{S}_{(c,d)},$ and an analytic function $\psi(z)$ that does not vanish on the real part of $\mathcal{S}_{(c,d)}$ such that locally uniformly in $z\in\mathcal{S}_{(c,d)},$

[TABLE]

where $t_{n}\rightarrow+\infty$ as $n\rightarrow\infty.$ Then we say that $X_{n}$ converges mod-$\phi$ on $\mathcal{S}_{(c,d)}$ with parameters $(t_{n})_{n\in{\mathbb{N}}}$ and limiting function $\psi.$

Assume that $\phi$ is a lattice distribution and the convergence is at speed $\mathcal{O}\left(\frac{1}{t^{\nu}_{n}}\right)$. Then Theorem 3.4. Feray, Meliot, and Nikeghbali (2013) says that for any $x\in{\mathbb{R}}$ in the interval $(\eta^{\prime}(c),\eta^{\prime}(d))$ and $\theta^{*}$ defined as $\eta^{\prime}(\theta^{*})=x,$ assume that $t_{n}x\in{\mathbb{N}},$ then,

[TABLE]

as $n\rightarrow\infty,$ where $F(x):=\sup_{\theta\in{\mathbb{R}}}\{\theta x-\eta(\theta)\}$ is the Legendre transform of $\eta(\cdot),$ and similarly, if $x\in{\mathbb{R}}$ is in the range of $(\eta^{\prime}(0),\eta^{\prime}(d)),$ then,

[TABLE]

as $n\rightarrow\infty,$ where $(a_{k})_{k=1}^{\infty},$ $(b_{k})_{k=1}^{\infty}$ are rational fractions in the derivatives of $\eta$ and $\psi$ at $\theta^{*},$ that can be computed as described in Remark 3.7. (Feray, Meliot, and Nikeghbali, 2013).

Then let’s recall Theorem 3.9. (Feray, Meliot, and Nikeghbali, 2013), which states the precise moderate deviation.

Consider a sequence $(X_{n})_{n\in\mathbb{N}}$ that converges mod-$\phi$, with a reference infinitely divisible law $\phi$ that is a lattice distribution. Assume $y=o((t_{n})^{1/6})$. Then,

[TABLE]

On the other hand, assuming $y\gg 1$ and $y=o((t_{n})^{1/2})$, if $x=\eta^{\prime}(0)+\sqrt{\eta^{\prime\prime}(0)/t_{n}}y$ and $h$ is the solution of $\eta^{\prime}(h)=x$, then

[TABLE]

And Corollary 3.13 (Feray, Meliot, and Nikeghbali, 2013) gives a more explicit form of Theorem 3.9. (Feray, Meliot, and Nikeghbali, 2013). If $y=o((t_{n})^{1/4})$, then one has

[TABLE]

More generally, if $y=o((t_{n})^{1/2-1/m})$, then one has

[TABLE]

Here we consider $N_{t}$ as a Cox process with Poisson shot noise intensity defined in (1), we assume throughout this paper that

• •

$\|g\|_{L^{1}}=\int_{0}^{\infty}g(t)dt<\infty$

• •

$g(t)=\mathcal{O}\left(\frac{1}{t^{\nu+2}}\right)$ as $t\to\infty$

By the definition of mod-$\phi$ convergence, the moment generating function of $N_{t}$ is finite, which implies the first assumption. We also need the second assumption for the convergence speed in mod-$\phi$ convergence.

Our main results for the precise large deviations and precise moderate deviations for the Cox process with Poisson shot noise intensity are stated as follows.

2.1 Precise Large Deviations

Definition 2.1**.**

We say that the sequence of random variables $(X_{n})_{n\in\mathbb{N}}$ converges mod-$\phi$ at speed $\mathcal{O}\left(\frac{1}{t^{\nu}_{n}}\right)$ if the difference of the two sides of Equation 2 can be bounded by $C_{K}\left(\frac{1}{t^{\nu}_{n}}\right)$ for any $z$ in a given compact subset $K$ of $\mathcal{S}_{(c,d)}$.

Theorem 2.2**.**

(i) For any $x>0,$ and $tx\in{\mathbb{N}},$ as $t\rightarrow\infty,$

[TABLE]

where for any $\theta\in\mathbb{C}$

[TABLE]

and

[TABLE]

which is analytic in $\theta$ for any $\theta\in\mathbb{C}$ and $I(x)$ is defined as

[TABLE]

and

[TABLE]

$\theta^{*}$* is the solution of $\eta^{\prime}(\theta^{*})=x$ and $(a_{k})_{k=1}^{\infty}$ are rational fractions in the derivatives of $\eta$ and $\psi$ at $\theta^{*}$.*

Note that $\theta^{*}$ is unique for each $x$ because it is easy to show that $\eta^{\prime\prime}(\theta)>0$.

(ii) For any $x>\nu+\rho\|g\|_{L_{1}},$ as $t\rightarrow\infty,$

[TABLE]

where $(b_{k})_{k=1}^{\infty}$ are rational fractions in the derivatives of $\eta$ and $\psi$ at $\theta^{*}$.

The key to prove main Theorem is to verify the mod-$\phi$ convergence. More precisely, we need to show the following three lemmas.

Lemma 2.3**.**

Y has an infinitely divisible distribution. Y is some random variable defined as

[TABLE]

Note that infinitely divisible is a limitation of the method of mod-$\phi$ convergence. Fortunately, the limiting distribution in the case of the Cox process with shot noise is indeed infinitely divisible.

Lemma 2.4**.**

For any $\theta\in\mathbb{C}$,

[TABLE]

is well-defined and analytic in $\theta,$ and

[TABLE]

as $t\rightarrow\infty,$ locally uniformly in $\theta$.

Lemma 2.5**.**

{$N_{t},t\geq 0$}* converges mod-$\phi$ at speed $\mathcal{O}\left(\frac{1}{t^{\nu}}\right)$ as $t\to\infty$.*

2.2 Precise Moderate Deviations

By Theorem 3.9. (Feray, Meliot, and Nikeghbali, 2013), we have the following theorem:

Theorem 2.6**.**

(i) For any $y=o(t^{1/6})$, as $t\to\infty$,

[TABLE]

where $\Psi(y):=\int_{y}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}dx$.

(ii) For any $y\gg 1$ and $y=o(t^{1/2})$, as $t\to\infty$,

[TABLE]

where $x^{*}=(\nu+\rho\|g\|_{L^{1}})+\frac{1}{\sqrt{t}}\sqrt{\nu+\rho\|g\|_{L^{1}}(\|g\|_{L^{1}}+1)}y$ and $\theta^{*}$ is the solution of $\eta^{\prime}(\theta^{*})=x^{*}$ and the notation $a\gg b$ means b is much less than a.

Remark 2.7**.**

Note that Theorem 2.6 (i) is an improvement of the classical central limit theorem because here we allow $y=o(t^{1/6})$ for $t\to\infty$. Theorem 2.6 (ii) is the moderate deviations.

By using Corollary 3.13 (Feray, Meliot, and Nikeghbali, 2013), we can get a more explicit form of Theorem 2.6.

Theorem 2.8**.**

(i) For any $y=o(t^{1/4})$, as $t\to\infty$,

[TABLE]

(ii) For any $y=o(t^{1/2-1/m})$, where $m\geq 3$, as $t\to\infty$,

[TABLE]

where $I(x)$ is defined in (12) and $\eta(\theta)$ is defined in (13).

3 Proofs

In this section, we give a proof of the main theorems and lemmas. The key idea to prove the main theorem is to apply the recently developed mod-$\phi$ convergence method.

3.1 Proofs of the results in Section 2.1

Proposition 3.1**.**

Assume that $N_{t}$ is Cox process with Poisson shot noise intensity defined in (1), then the moment generating function of $N_{t}$ is

[TABLE]

Proof.

By the definition of a Cox process with Poisson shot noise intensity and Fubini’s theorem, we have

[TABLE]

∎

Remark 3.2**.**

From (22), we have

[TABLE]

Then, by Gärtner-Ellis theorem (see for details (Bordenave and Torrisi, 2007)), we can say that $(\frac{N_{t}}{t}\in\cdot)$ satisfies the large deviation principle with the good rate function

[TABLE]

Proof of Lemma 2.3.

It suffice to show that

[TABLE]

where Y is an infinitely divisible random variable. To this end, Y can be interpreted as $Y=X+Z$. Here $X$ is a Poisson random variable with parameter $\nu$, $Z=\sum_{i=1}^{R}Z_{i}$ is a compound Poisson random variable, $R$ is a Poisson random variable with parameter $\rho$, $Z_{i}$ are i.i.d. Poisson random variables with parameter $\|g\|_{L^{1}}$, also, $X$ and $Z$ are independent thus

[TABLE]

Therefore, $Y$ has an infinitely divisible distribution. ∎

Proof of Lemma 2.4.

We first consider

[TABLE]

Using (22), we have

[TABLE]

To prove this, it suffices to show that

[TABLE]

as $t\rightarrow\infty,$ locally uniformly in $\theta\in\mathbb{C}.$ It is equivalent to show that for any compact set $K\subset\mathbb{C}$, $\forall\epsilon>0,\exists N$ s.t. when $t>N$,

[TABLE]

for $\theta\in K,$ and we have

[TABLE]

by the mean value theorem, $\xi(u)\in\left(\int_{0}^{u}g(s)ds,\int_{0}^{\infty}g(s)ds\right)$.

[TABLE]

where \hat{C}_{K}=\sup_{\theta\in K}\Bigl{\{}(e^{\theta}-1)e^{(e^{\theta}-1)\int_{0}^{\infty}g(s)ds}\Bigr{\}}. According to our assumption, $g(t)=\mathcal{O}\left(\frac{1}{t^{\nu+2}}\right)$, so $\left|\int_{t}^{\infty}\int_{u}^{\infty}g(s)dsdu\right|=\mathcal{O}\left(\frac{1}{t^{\nu}}\right)$ as $t\to\infty$.

So we have for $\epsilon^{\prime}=\epsilon/\hat{C}_{K}$ there exists $N$ such that if $t>N$,

[TABLE]

then for $t>N$,

[TABLE]

so we have proved (29).

Thus, we conclude that

[TABLE]

as $t\rightarrow\infty,$ locally uniformly in $\theta\in\mathbb{C},$ where

[TABLE]

Hence, $\varphi(\theta)$ is well-defined and is analytic in $\theta.$ ∎

Proof of lemma 2.5.

By definition 2.1, it suffices to show that

[TABLE]

as $t\to\infty$. Here $G(u)=e^{(e^{\theta}-1)\int_{0}^{u}g(s)ds}-e^{(e^{\theta}-1)\int_{0}^{\infty}g(s)ds}$. According to (30) and (31),

[TABLE]

as $t\to\infty.$ Here $\xi^{\prime}\in(t,\infty)$. Thus we have proved Lemma 2.5.

∎

Completion of the Proof of Theorem 2.2.

From Lemma 2.3 and Lemma 2.4, we have established the mod-$\phi$ convergence. Hence, by Theorem 3.4. (Feray, Meliot, and Nikeghbali, 2013), for any $x>0,$ and $tx\in{\mathbb{N}},$

[TABLE]

and for any $x>\nu+\rho\|g\|_{L_{1}},$

[TABLE]

where $I(x)$ is defined in (25). This completes the proof of Theorem 2.2. ∎

3.2 Proofs of the results in Section 2.2

Since we have established the mod-$\phi$ convergence and $N_{t}$ is a lattice distribution, the proof of Theorem 2.6 and Theorem 2.8 follows from Theorem 3.9 and Corollary 3.13 in (Feray, Meliot, and Nikeghbali, 2013), respectively.

**Acknowledgements

**Youngsoo Seol is grateful to the support from the Dong-A University research grant.