Prelimit and limit generalizations of the Pollaczek–Khinchin formula
Author:
D. V. Gusak
Translated by:
S. V. Kvasko
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 \[ \xi (t)=at+S(t), \quad a<0, \qquad S(t)=\sum _{k\leq \nu (t)}\xi _k, \quad \xi _k>0, \] 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
- A. A. Borovkov, Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0315800
- G. P. Klimov, Stokhasticheskie sistemy obsluzhivaniya, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0207064
- Søren Asmussen, Ruin probabilities, Advanced Series on Statistical Science & Applied Probability, vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1794582
- 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)
- 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, DOI https://doi.org/10.1023/A%3A1020557215381
- 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
References
- 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)
- G. P. Klimov, Stochastic Queueing Systems, Nauka, Moscow, 1966. (Russian) MR 0207064 (34:6880)
- S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000. MR 1794582 (2001m:62119)
- 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)
- 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)
- 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
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