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On some first-crossing-time probabilities for a two-dimensional random walk with correlated components

Part of: Stochastic processes Combinatorics

Published online by Cambridge University Press:  01 July 2016

A. Di Crescenzo ,
V. Giorno  and
A. G. Nobile
A. Di Crescenzo*
Affiliation:
University of Naples
V. Giorno*
Affiliation:
University of Salerno
A. G. Nobile*
Affiliation:
University of Udine
*
∗ Postal address: Dipartimento di Matematica e Applicazioni, University of Naples, via Cintia, 80126 Naples, Italy.
∗∗ Postal address: Dipartimento di Informatica e Applicazioni, University of Salerno, via S. Allende, 84081 Baronissi (SA), Italy. Correspondence should be addressed to this author.
∗∗∗ Postal address: Dipartimento di Matematica e Informatica, University of Udine, via Zanon 6, 33100 Udine, Italy.
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Abstract

For a two-dimensional random walk {X (n) = (X(n)1, X(n)2)T, n ∈ ℕ0} with correlated components the first-crossing-time probability problem through unit-slope straight lines x2 = x1 - r(r = 0,1) is analysed. The p.g.f.'s for the first-crossing-time probabilities are expressed as solutions of a fourth-degree algebraic equation and are then exploited to obtain the first-crossing-time probabilities. Several additional results, including the mean first-crossing time and the probability of ultimate crossing, are also given.

Keywords

UNIT-SLOPE LINESULTIMATE CROSSINGMEAN CROSSING TIME

MSC classification

Secondary: 05A15: Exact enumeration problems, generating functions 92D25: Population dynamics (general) 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 441 - 454
Copyright
Copyright © Applied Probability Trust 1992 

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References

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