We show that gamma distributions, generalized positive Linnik distributions, S2 distributions are fixed points of Poisson shot noise transforms. The corresponding response functions are identified via their inverse functions except for some special cases when those can be obtained explicitly. As a by-product, it is proven that log-convexity of the response function is not necessary for selfdecomposability of non-negative Poisson shot noise distribution. Some attention is given to perpetuities of a rather special type which are closely related to our model. In particular, we study the problem of their existence and uniqueness.
Poisson shot noise processes are natural generalizations of compound Poisson processes that have been widely applied in insurance, neuroscience, seismology, computer science and epidemiology. In this paper we study sharp deviations, fluctuations and the stable probability approximation of Poisson shot noise processes. Our achievements extend, improve and complement existing results in the literature. We apply the theoretical results to Poisson cluster point processes, including generalized linear Hawkes processes, and risk processes with delayed claims. Many examples are discussed in detail.
Distributional fixed points of a Poisson shot noise transform (for nonnegative, nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than -1. The uniqueness of fixed points is also discussed. Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.
We investigate the regularity of shot noise series and of Poisson integrals. We give conditions for the absolute continuity of their law with respect to Lebesgue measure and for their continuity in total variation norm. In particular, the case of truncated series in adressed. Our method relies on a disintegration of the probability space based on a mere conditioning by the first jumps of the underlying Poisson process.
Shot-noise and fractional Poisson processes are instances of filtered Poisson processes. We here prove Girsanov theorem for this kind of processes and give an application to an estimate problem.
In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the result available on fractional Poisson processes in several directions.
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-$ϕ$ convergence method.
We consider a risk model with a counting process whose intensity is a Markovian shot-noise process, to resolve one of the disadvantages of the Cramér-Lundberg model, namely the constant jump intensity of the Poisson process. Due to this structure, we can apply the theory of PDMPs on a multivariate process containing the intensity and the reserve process, which allows us to identify a family of martingales. Eventually, we use change of measure techniques to derive an upper bound for the ruin probability in this model. Exploiting a recurrent structure of the shot-noise process, even the asymptot…
We consider the maximum process of a random walk with additive independent noise in form of $\max_{i=1,\dots,n}(S_i+Y_i)$. The random walk may have dependent increments, but its sample path is assumed to converge weakly to a fractional Brownian motion. When the largest noise has the same order as the maximal displacement of the random walk, we establish an invariance principle for the maximum process in the Skorohod topology. The limiting process is the maximum process of the fractional Brownian notion with shot noise generated by Poisson point processes.
Shot noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory and in the engineering sciences. In this work we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot noise processes. The result covers previously known large deviation results for one dimensional state-independent shot noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate conver…
Let $μ_{g}$ and $μ_{p}$ denote the Gaussian and Poisson measures on ${\Bbb R}$, respectively. We show that there exists a unique measure $\widetildeμ_{g}$ on ${\Bbb C}$ such that under the Segal-Bargmann transform $S_{μ_g}$ the space $L^2({\Bbb R},μ_g)$ is isomorphic to the space ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$ of analytic $L^2$-functions on ${\Bbb C}$ with respect to $\widetildeμ_{g}$. We also introduce the Segal-Bargmann transform $S_{μ_p}$ for the Poisson measure $μ_{p}$ and prove the corresponding result. As a consequence, when $μ_{g}$ and $μ_{p}$ have the same variance, $L^2({\Bb…
We consider a Markov chain on the space of (countable) partitions of the interval [0,1], obtained first by size biased sampling twice (allowing repetitions) and then merging the parts (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). We prove a conjecture of Vershik stating that the Poisson-Dirichlet law with parameter theta=1 is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric, and representation-theoretic arguments.
We study infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the arrival rate is given by shot noise. A shot-noise rate emerges as a natural model, if the arrival rate tends to display sudden increases (or: shots) at random epochs, after which the rate is inclined to revert to lower values. Exponential decay of the shot noise is assumed, so that the queueing systems are amenable for analysis. In particular, we perform transient analysis on the number of customers in the queue jointly with the value of the driving shot-noise proc…
The following random recurrency: $$ W_{n+1} = U_{n+1} ( W_n + Λ_{n+1} ) $$ where $W_{-1}=0.$ is known to be associated with the shot noise : $$ W_{t} = \sum_{0<t_k<t} Λ_{k} e^{-(t-t_k)} $$ where the $t_k$ are the dates of a Poisson process, the $ U_i ;\ i = 0,1,... $ are independent uniform variables. This can be extended to the triggered shot noise : $$ W_{t} = \sum_{0<t_{lk}<t} Λ_{lk} e^{-(t-t_{lk})} , l=1,2,3,... $$ The random amplitudes $Λ_i,\ i = 0,1,... $ are positive Dufresne independent variables or also even density variables which characteristic function given by generalized hypergeo…
In this paper we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random…
We consider a class of probability measures $μ_{s,r}^α$ which have explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify $μ_{s,2}^α$ as a free compound Poisson law with Lévy measure a monotone $α$-stable law. This implies the free infinite divisibility of $μ_{s,2}^α$. Moreover, when symmetric or positive, $μ_{s,2}^α$ has a representation as the free multiplication of a free Poisson law and a monotone $α$-stable law. We also investigate the free infinite divisibility of $μ_{s,r}…
This article studies the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, it is shown that the shot-noise field can be scaled suitably to have a $α$-stable limit, intensity of the underlying point process goes to infinity. It is also shown that the finite dimensional distributions of the limiting random field have i.i.d. stable random components. We hence propose to call this limte the $α$- stable white noise field. Analogous results are also obtained for the e…
