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.”
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This work was supported by the Russian Foundation for Basic Research (grant no. 20-11-00646 a).
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Translated by I. Tselishcheva
APPENDIX. A TAUBERIAN THEOREM
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 Fλ(x) be its distribution function; and let Lλ(s) be its Laplace transform. If the function \(\ell \) changes slowly at zero and 0 \(\leqslant \) p < ∞, then each of the relations
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,
The function \(\ell \)(t) is said to vary slowly at zero if \(\ell \left( {\frac{1}{t}} \right)\) slowly varies at infinity.
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Rusakov, O.V., Ragozin, R.A. Extremes of PSI-Processes and Gaussian Limits of Their Normalized Independent Identically Distributed Sums. Vestnik St.Petersb. Univ.Math. 55, 186–191 (2022). https://doi.org/10.1134/S106345412202011X
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DOI: https://doi.org/10.1134/S106345412202011X