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On crossing times for multidimensional walks with skip-free components

Part of: Markov processes Combinatorial probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

K. A. Borovkov
K. A. Borovkov*
Affiliation:
Steklov Mathematical Institute
*
∗Postal address: Steklov Mathematical Institute, Vavilov st. 42, 117966 Moscow GSP–1, Russia.
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Abstract

The paper deals with processes in (or ), one of whose components is skip-free. We obtain identities for distributions of hitting times for the components of the process generalizing the well-known one for the one-dimensional case. These relations reflect the fact that in this case spatial and time coordinates play, in some sense, symmetric roles. They turn out to be useful for solving several problems. For example, they allow us to find the distribution of the number of jumps of the process, which fall in a fixed set before the skip-free component of the process hits a fixed level. Examples are given showing how our results can be applied to models in branching processes, queueing, and risk theory.

Keywords

SKIP-FREE RANDOM WALKSHITTING TIMES

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J55: Local time and additive functionals 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 991 - 1006
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Supported by the Alexander von Humboldt Foundation. This work was done while the author was visiting the Carl von Ossietzky University, Oldenburg.

References

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