The bivariate gamma distribution and the random walk problem

Abstract

The relation between the random walk problem in one dimension and the joint distribution of two correlated gamma variates of parameter ½ is firstly established. It is shown that the two variates which represent respectively the squares of distances from the origin (properly normalised) after P steps and P+Q (=N) steps in one dimension are a pair of correlated gamma variates with parameter ½. This is used as a heuristic principle to set up similar bivariate gamma distribution with parameters 1 by examining the corresponding problem of random walk in two dimensions. Finally, the results are extended to the case of generaln-dimensional random walk where it is shown to lead to the bivariate gamma distribution with parametern/2.