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

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 $\{x_i\}$ has been called $m$-dependent [3] if $| i - j | > m$ implies that $x_i$ and $x_j$ 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.