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Theory of Probability and Mathematical Statistics

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Exit time functionals for integer-valued Poisson processes


Author: D. V. Gusak
Translated by: V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 68 (2003).
Journal: Theor. Probability and Math. Statist. 68 (2004), 27-39
MSC (2000): Primary 60G50, 60J70
DOI: https://doi.org/10.1090/S0094-9000-04-00603-9
Published electronically: June 10, 2004
MathSciNet review: 2000392
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Abstract | References | Similar Articles | Additional Information

Abstract: The joint distribution of all exit time functionals is studied in this paper for a fixed level $x$ and integer-valued compound Poisson processes. An exact formula for the distributions of these functionals is obtained in the case of semicontinuous processes. Limit relations are obtained for the distributions of the exit time functionals for $x=0$ or as $x\to\infty$.


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

  • 1. D. V. Gusak, The joint distribution of the time and magnitude of the first overshoot for homogeneous processes with independent increments, Teor. Verojatnost. i Primenen. 14 (1969), 15–23 (Russian, with English summary). MR 0245083
  • 2. 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, https://doi.org/10.1023/A:1020557215381
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  • 4. D. V. Gusak and A. I. Tureniyazova, Distribution of some functionals for lattice Poisson processes on a Markov chain, Asymptotic methods in the investigation of stochastic models (Russian), Acad. Sci. Ukrain. SSR, Inst. Math., Kiev, 1987, pp. 21–27, 143 (Russian). MR 943353
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  • 6. D. V. Gusak, On a generalized semicontinuous integer-valued Poisson process with reflection, Teoriya Imovir. ta Matem. Statist. 59 (1998), 41-46; English transl., Theor. Probability and Math. Statist. 59 (1999), 41-46.
  • 7. -, The factorization method in boundary problems for homogeneous processes with independent increments, Distribution of Some Functionals for Processes with Independent Increments, Preprint 85-43, Institute of Mathematics, Academy of Sciences of Ukrainian SSR, Kiev, 1985, pp. 3-42. (Russian)

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Additional Information

D. V. Gusak
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka Street 3, Kyiv 01601, Ukraine
Email: random@imath.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-04-00603-9
Received by editor(s): February 18, 2002
Published electronically: June 10, 2004
Article copyright: © Copyright 2004 American Mathematical Society