4 - Random Walk and Rank Order Statistics

Abstract

This chapter discusses the applications of path counting to distributional problems in random walks and statistical nonparametric inferences. Discrete random walks by their very nature, especially when the destination at a given time is fixed, can be represented by paths and thus, any probability distribution of characteristics defined on a random walk involves the enumeration of paths with restrictions constrained by the characteristics. Because of a very interesting property, rank order statistics arising in non-parametric inferences are directly related to paths. Using this relation, which is called Gnedenko's technique, problems of the distribution of these statistics are discussed. Because of the correspondence with paths, one may observe that a particular distributional problem can be formulated in terms of either a restricted random walk or rank order statistics. It is well known that random walk models serve as a first approximation to the theory of Brownian motion. Random walks have also helped in finding the distribution of mainly rank order statistics that arise in two-sample problems.