On a probability problem connected with Railway traffic

Lajos Takács

International Journal of Stochastic Analysis, 1991, vol. 4, 1-27

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.

Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:363025

DOI: 10.1155/S1048953391000011

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