Extreme Values in Uniformly Mixing Stationary Stochastic Processes

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables, and write $Z_n$ for the maximum of $X_1, X_2, \cdots X_n$. Then there are two well known theorems concerning the limiting behaviour of the distribution of $Z_n$. (See, for example, Gumbel [7].) Firstly, if for some sequence of pairs of numbers $a_n, b_n$, the quantities $a^{-1}_n(Z_n - b_n)$ have a non-degenerate limiting distribution as $n \rightarrow \infty$, then this limit must take one of three forms. Secondly, if $c_n = c_n(\xi)$ is defined by $P\lbrack X > c_n\rbrack \leqq \xi/n \leqq P\lbrack X \geqq c_n\rbrack$, then $P\lbrack Z_n \leqq c_n\rbrack$ tends to $e^{-\xi}$ as $n \rightarrow \infty$. Suppose now that we drop the assumption of independence of the $X_j$, and require instead that the sequence $\{X_n\}$ be a stationary stochastic process: then it might be expected that similar results will hold, at least if $X_i$ and $X_j$ are nearly independent when $|i - j|$ is large. In Section 2 it will be shown that, if the process $\{X_n\}$ is uniformly mixing, then the only possible non-degenerate limit laws of $a^{-1}_n(Z_n - b_n)$ are just those that occur in the case of independence, and that the only possible limit laws of $P\lbrack Z_n \leqq c_n\rbrack$ are of the form $e^{-k\xi}, k$ being some positive constant not greater than one. The uniform mixing property is rather strong at first sight. It is however clear that some restriction is necessary, at least for example to ergodic processes, and the parallel which exists to a certain extent with normed sums of random variables suggests that uniform mixing hypotheses may be appropriate. (Cf. Rosenblatt [9]). In the independent case converse results hold, as we have already observed for the second problem, giving necessary and sufficient conditions for the existence of the limits. We give some results concerning this problem in Section 3, but they are not altogether satisfactory. Berman ([2] and especially [3]) has investigated the same problem, under somewhat different conditions. His results for the Gaussian case in [3], however, include those which can be obtained from Lemmas 1 and 2 of the present paper, and in consequence we do not reproduce them here. The second problem mentioned above was considered by Watson [10] for $m$-dependent stationary processes, and his paper did in fact suggest the present investigations. His results are contained in Section 3. Certain results were announced without proof by Chibisov [4], but these appear not to overlap our results.