The Annals of Mathematical Statistics

Pairs of One Dimensional Random Walk Paths

C. H. Raifaizen

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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.

Article information

Source
Ann. Math. Statist., Volume 43, Number 6 (1972), 2095-2098.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177690891

Digital Object Identifier
doi:10.1214/aoms/1177690891

Mathematical Reviews number (MathSciNet)
MR356246

Zentralblatt MATH identifier
0248.60047

JSTOR
links.jstor.org

Citation

Raifaizen, C. H. Pairs of One Dimensional Random Walk Paths. Ann. Math. Statist. 43 (1972), no. 6, 2095--2098. doi:10.1214/aoms/1177690891. https://projecteuclid.org/euclid.aoms/1177690891


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