Random Walk Processes and their Applications in Order Statistics

This paper is concerned with two stochastic processes, namely, a Bernoulli excursion and a tied-down random walk. Three random variables are defined for these processes, each variable representing the area of a random set determined by one of the processes. The aim is to find the distributions and the moments of these random variables and to determine their asymptotic behavior. The results derived for random walks are applied to the theory of order statistics to determine the asymptotic behavior of the moments and the distributions of two statistics which measure the deviation between two empirical distribution functions.