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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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  • 1. A. A. Borovkov, \cyr Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya., Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0315800
    A. A. Borovkov, Stochastic processes in queueing theory, Springer-Verlag, New York-Berlin, 1976. Translated from the Russian by Kenneth Wickwire; Applications of Mathematics, No. 4. MR 0391297
  • 2. G. P. Klimov, \cyr Stokhasticheskie sistemy obsluzhivaniya, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0207064
  • 3. Søren Asmussen, Ruin probabilities, Advanced Series on Statistical Science & Applied Probability, vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1794582
  • 4. D. V. Gusak, Limit problems for processes with independent increments in the risk theory, Proceedings of the Institute of Mathematics, National Academy of Science of Ukraine, vol. 67, Kiev, 2007. (Ukrainian)
  • 5. D. V. Gusak, Distribution of overshoot functionals of a semicontinuous homogeneous process with independent increments, Ukraïn. Mat. Zh. 54 (2002), no. 3, 303–322 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 3, 371–397. MR 1952790,
  • 6. A. V. Skorohod, Random processes with independent increments, Mathematics and its Applications (Soviet Series), vol. 47, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the second Russian edition by P. V. Malyshev. MR 1155400

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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