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The fundamental matrix for a certain random walk

Part of: Basic linear algebra

Published online by Cambridge University Press:  14 July 2016

Howard M. Taylor
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Abstract

Consider the random walk {Sn} whose summands have the distribution P(X=0) = 1-(2/π), and P(X = ± n) = 2/[π(4n2−1)], for n ≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean time xk for the random walk starting from S0 = k to exit the interval. The explicit formula yields the limiting behavior of xk as N → ∞ with k fixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By letting N → ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting from k = 0 results. The distributions for some related random variables are also discovered.

Applications to stress concentration calculations in discrete lattices are briefly reviewed.

Keywords

Random walkToeplitz matrixWiener–Hopf

MSC classification

Secondary: 15A23: Factorization of matrices
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 320 - 333
Copyright
Copyright © Applied Probability Trust 1999 

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