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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


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.


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C. H. Raifaizen. "Pairs of One Dimensional Random Walk Paths." Ann. Math. Statist. 43 (6) 2095 - 2098, December, 1972.


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

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

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 6 • December, 1972
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