We prove the Simons-Johnson theorem for the sums $S_n$ of $m$-dependent random variables, with exponential weights and limiting compound Poisson distribution $\CP(s,λ)$. More precisely, we give sufficient conditions for $\sum_{k=0}^\infty\ee^{hk}\ab{P(S_n=k)-\CP(s,λ)\{k\}}\to 0$ and provide an estimate on the rate of convergence. It is shown that the Simons-Johnson theorem holds for weighted Wasserstein norm as well. %limiting sum of two Poisson variables defined on %different lattices. The results are then illustrated for $N(n;k_1,k_2)$ and $k$-runs statistics.
In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between α-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorith…
This note aims at presenting several new theoretical results for the compound Poisson point process, which follows the work of Zhang \emph{et al.} [Insurance~Math.~Econom.~59(2014), 325-336]. The first part provides a new characterization for a discrete compound Poisson point process (proposed by {Acz{é}l} [Acta~Math.~Hungar.~3(3)(1952), 219-224]), it extends the characterization of the Poisson point process given by Copeland and Regan [Ann.~Math.~(1936): 357-362]. Next, we derive some concentration inequalities for discrete compound Poisson point process (negative binomial random variable wit…
Given data from a Poisson point process with intensity $(x,y) \mapsto n \mathbf{1}(f(x)\leq y),$ frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone and piecewise constant support boundaries and adapts to Hölder smooth boundaries with smoothness index at most one. We then derive a non-standard Bernstein-von Mises result for a compound Poisson process prior and a function space with increasing paramet…
A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as conf…
The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter depending on $x$.
A compound Poisson process whose jump measure and intensity are unknown is observed at finitely many equispaced times. We construct a purely data-driven estimator of the Lévy density $ν$ through the spectral approach using general Calderon--Zygmund integral operators, which include convolution and projection kernels. Assuming minimal tail assumptions, it is shown to estimate $ν$ at the minimax rate of estimation over Besov balls under the losses $L^p(\mathbb{R})$, $p\in[1,\infty]$, and robustly to the observation regime (high- and low-frequency). To achieve adaptation in a minimax sense, we us…
For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the …
We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erdös (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and bin…
We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form $$Y_t = \sum_{k=1}^{N(t)} Z_k,~~~ t \ge 0,$$ where $N(t)$ is a standard Poisson process of intensity $λ$, and $Z_k$ are drawn i.i.d.~from jump measure $μ$. A high-dimensional wavelet series prior for the Lévy measure $ν= λμ$ is devised and the posterior distribution arises from observing discrete samples $Y_Δ, Y_{2Δ}, \dots, Y_{nΔ}$ at fixed observation distance $Δ$, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior…
