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Wald's identity and absorption probabilities for two-dimensional random walks

Published online by Cambridge University Press:  24 October 2008

V. D. Barnett
V. D. Barnett
Affiliation:
University of Birmingham
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Summary

Suppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), where

and we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)

where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 3 , July 1965 , pp. 747 - 762
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

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