Vol. 99, No. 1, 1982

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ISSN: 0030-8730
An application of orthogonal polynomials to random walks

Thomas Alva Whitehurst

Vol. 99 (1982), No. 1, 205–213
Abstract

If Xn is a simple random walk on the nonnegative integers with transition probabilities Pij(k) = Pr{Xn+k = jXn = i}, then Pij(k) has an integral representation in terms of a family of orthogonal polynomials and the associated probability distribution function F(x) for these polynomials. The relationship between the distribution F, the family of polynomials and the random walk Xn is studied. Necessary and sufficient conditions for the support of F to be contained in [0,1] are given.

Mathematical Subject Classification 2000
Primary: 60J15
Secondary: 60J10
Milestones
Received: 4 February 1980
Published: 1 March 1982
Authors
Thomas Alva Whitehurst