Extremes of PSI-Processes and Gaussian Limits of Their Normalized Independent Identically Distributed Sums

Abstract

By a Poisson stochastic index process (PSI-process), is meant a continuous-time random process obtained by the discrete-time randomization of a random sequence. The case when this randomization is generated by a doubly stochastic Poisson process, i.e., a Poisson process with a random intensity, is considered. Under the condition of existence of the second moment, stationary PSI-processes have a covariance coinciding with the Laplace transform of a random intensity. In this paper, distributions for extremes of one PSI-process that are expressed in terms of the Laplace transform of a random intensity are obtained. The second problem that is solved here is the convergence of the maximum Gaussian limit for normalized sums of independent identically distributed stationary PSI-processes. Necessary and sufficient conditions imposed on the random intensity for a suitably centered and normalized maximum of this Gaussian limit to converge in distribution to the double exponential law are found. For this purpose, we essentially use a Tauberian theorem in Feller’s form and the results of the monograph by Leadbetter et al. (1983), “Extremes and Relative Properties of Random Sequences and Processes.”

APPENDIX. A TAUBERIAN THEOREM

Let us present the Tauberian theorem from the book [9] (Theorem 1.7.1') with the notation adapted for our paper.

Theorem 3. Let λ be a nonnegative random variable, let F_λ(x) be its distribution function; and let L_λ(s) be its Laplace transform. If the function $\ell$ changes slowly at zero and 0 $\leqslant$ p < ∞, then each of the relations

$${{L}_{\lambda }}(s)\sim c{{s}^{{ - p}}}\ell (s),\quad s \to \infty ,$$

$${{F}_{\lambda }}(x)\sim \frac{c}{{\Gamma (p + 1)}}{{x}^{p}}\ell (1{\text{/}}x),\quad x \to 0,$$

implies another for c $\geqslant$ 0.

Note that the book [9] by Bingham et al. gives Theorem 1.7.1 with reference to Tauberian theorems from the famous monograph by Feller [1]. We recall below the definition of a slowly varying function.

Definition 2. A positive function $\ell$ is called slowly varying at infinity if, for every fixed x > 0,

$$\frac{{\ell (tx)}}{{\ell (t)}} \to 1,\quad t \to \infty .$$

The function $\ell$(t) is said to vary slowly at zero if $\ell \left( {\frac{1}{t}} \right)$ slowly varies at infinity.