On a probability problem connected with Railway traffic. (1991) - Takács, Lajos
Abstract : Let F n ( x ) and G n ( 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 θ n by θ n / n = ∫ 0 1 [ F n ( x ) − G n ( x ) ] d V ( x ) − min 0 ≤ x ≤ 1 [ F n ( x ) − G n ( x ) ] . In this paper the limits of E { ( θ n / 2 n ) r } ( r = 0, 1, 2, … ) and P { θ n / 2 n ≤ x } are determined for n → ∞ . The problem of finding the asymptotic behavior of the moments and the distribution of θ n as n → ∞ has arisen in a study of the fluctuations of the inventory of locomotives in a randomly chosen railway depot.