Extreme Values in Samples from $m$-Dependent Stationary Stochastic Processes

Abstract

The limiting distributions for the order statistics of n successive observations in a sequence of independent and identically distributed random variables are shown to hold also when the sequence is generated by a stationary stochastic process of a certain moving average type. A sequence of random variables {xi} has been called m-dependent [3] if $| i - j| > m$ implies that xi and xj are independent. If the variables in a strictly stationary sequence are m-dependent and have a finite upper bound to their range of variation, the largest in a sample of n successive members tends with probability one to this upper bound. This is a simple extension of Dodd's results [1] for the case of independence. The following theorem shows that when this upper bound is infinite, the asymptotic distribution of the largest in such a sample is the same as in the case of independence.