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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
DOI: https://doi.org/10.1090/S0094-9000-2010-00804-0
Published electronically: August 20, 2010
MathSciNet review: 2541950
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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.


References [Enhancements On Off] (What's this?)

  • 1. A. A. Borovkov, Stochastic Processes in Queueing Theory, Nauka, Moscow, 1972; English transl., Springer-Verlag, New York-Heidelberg-Berlin, 1976. MR 0315800 (47:4349); MR 0391297 (52:12118)
  • 2. G. P. Klimov, Stochastic Queueing Systems, Nauka, Moscow, 1966. (Russian) MR 0207064 (34:6880)
  • 3. S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000. MR 1794582 (2001m:62119)
  • 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 overjump functionals of semicontinuous homogeneous processes with independent increments, Ukrain. Matem. Zh. 54, no. 3, 303-322; English transl. in Ukr. Math. J. 54 (2002), no. 3, 371-379. MR 1952790 (2003j:60067)
  • 6. A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964; English transl., Kluwer Academic Publishers, Dordrecht, 1991. MR 1155400 (93a:60114)

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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
Email: random@imath.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2010-00804-0
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

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