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December, 1972 Pairs of One Dimensional Random Walk Paths
C. H. Raifaizen
Ann. Math. Statist. 43(6): 2095-2098 (December, 1972). DOI: 10.1214/aoms/1177690891

Abstract

We first show a way of constructing a path of length $2n$ from a pair of paths of length $n$ by means of which one may arrive at many results on pairs of paths of length $n$, simply by examining properties of paths of length $2n$. Secondly, for two random walk paths of length $n, A$ and $B$, with vertical coordinates $A(i)$ and $B(i)$ respectively, at times $i = 0,1,\cdots, n$, and such that for some $m A(m) > B(m)$ but $A(i) = B(i)$ when $i < m$, we define $d_{A,B}(i) = \frac{1}{2}(A(i) - B(i))$. For obvious reasons $A(i) - B(i)$ is always even, which incidentally, implies that the intersection of two paths are points with integral coordinates. We find that $d_{A,B}$ can be graphed against time by a three-valued random walk path, i.e. a path which may have horizontal steps. Questions about the pair consisting of $A$ and $B$ may then be answered by observing the path described by $d_{A,B}$. Results in the theory of three-valued random walk paths can thus be translated into results about pairs of random walk paths of equal length.

Citation

C. H. Raifaizen. "Pairs of One Dimensional Random Walk Paths." Ann. Math. Statist. 43 (6) 2095 - 2098, December, 1972. https://doi.org/10.1214/aoms/1177690891

Information

Published: December, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0248.60047
MathSciNet: MR356246
Digital Object Identifier: 10.1214/aoms/1177690891