Pairs of One Dimensional Random Walk Paths
Raifaizen, C. H.
Ann. Math. Statist., Tome 43 (1972) no. 6, p. 2095-2098 / Harvested from Project Euclid
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.
Publié le : 1972-12-14
Classification: 
@article{1177690891,
     author = {Raifaizen, C. H.},
     title = {Pairs of One Dimensional Random Walk Paths},
     journal = {Ann. Math. Statist.},
     volume = {43},
     number = {6},
     year = {1972},
     pages = { 2095-2098},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177690891}
}
Raifaizen, C. H. Pairs of One Dimensional Random Walk Paths. Ann. Math. Statist., Tome 43 (1972) no. 6, pp.  2095-2098. http://gdmltest.u-ga.fr/item/1177690891/