Journal of Applied Mathematics and Stochastic Analysis
Volume 4 (1991), Issue 1, Pages 1-27
doi:10.1155/S1048953391000011

On a probability problem connected with Railway traffic

Lajos Takács1,2

1Case Western Reserve University, Cleveland, Ohio, USA
22410 Newbury Drive, Cleveland Heights, 44118, OH, USA

Received 1 December 1990; Revised 1 January 1991

Copyright © 1991 Lajos Takács. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Fn(x) and Gn(x) be the empirical distribution functions of two independent samples, each of size n, in the case where the elements of the samples are independent random variables, each having the same continuous distribution function V(x) over the interval (0,1). Define a statistic θn by θn/n=∫01[Fn(x)−Gn(x)]dV(x)−min0≤x≤1[Fn(x)−Gn(x)]. In this paper the limits of E{(θn/2n)r}(r=0,1,2,…) and P{θn/2n≤x} are determined for n→∞. The problem of finding the asymptotic behavior of the moments and the distribution of θn as n→∞ has arisen in a study of the fluctuations of the inventory of locomotives in a randomly chosen railway depot.