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Theory of Probability and Mathematical Statistics

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Prelimit and limit generalizations of the Pollaczek-Khinchin formula

Author: D. V. Gusak
Translated by: S. V. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 80 (2009).
Journal: Theor. Probability and Math. Statist. 80 (2010), 37-46
MSC (2000): Primary 60G50; Secondary 60K10
Published electronically: August 20, 2010
MathSciNet review: 2541950
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Abstract | References | Similar Articles | Additional Information

Abstract: The moment generating function of the nondegenerate distribution of the maximum $ \xi^+=\sup_{0\leq t<\infty}\xi(t)$ of a compound Poisson process

$\displaystyle \xi(t)=at+S(t), \quad a<0, \qquad S(t)=\sum_{k\leq\nu(t)}\xi_k, \quad \xi_k>0, $

where $ \nu(t)$ is a simple Poisson process with intensity $ \lambda>0$, is determined via the well-known Pollaczek-Khinchin formula if $ m=\mathsf{E}\xi(1)<0$.

We obtain a prelimit generalization of this formula that determines the Laplace-Carson transform of the moment generating function of the maximum $ \xi^+(t)=\sup_{0\leq t'\leq t}\xi(t')$, $ 0<t<\infty$, and the moment generating function of $ \xi^+=\xi^+(\infty)$ under the assumption that $ m<0$ for homogeneous processes $ \xi(t)$ with independent increments and of bounded variation. Relationships of a different type between characteristic functions of $ \xi^+(\theta_s)$ $ (\mathsf{P}\{\theta_s>t\}=e^{-st}, s,t>0)$ and of $ \xi^+$ are also obtained by using earlier results presented by the author.

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Additional Information

D. V. Gusak
Affiliation: 252601, Institute of Mathematics, National Academy of Sciences of Ukraine, Tere- shchenkivs’ka Street, 3, Kiev 01004, Ukraine

Keywords: Semicontinuous compound Poisson processes, semicontinuous homogeneous processes with independent increments, Pollaczek–Khinchin formula and its generalizations
Received by editor(s): February 27, 2009
Published electronically: August 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society