A Convergence Theorem for Extreme Values from Gaussian Sequences

Abstract

Let $\{X_n, n = 0, \pm 1, \pm 2, \cdots\}$ be a stationary Gaussian stochastic process with means zero, variances one, and covariance sequence $\{r_n\}$. Let $M_n = \max \{X_1, \cdots, X_n\}$ and $S_n =$ second largest $\{X_1, \cdots, X_n\}$. Limit properties are obtained for the joint law of $M_n$ and $S_n$ as $n$ approaches infinity. A joint limit law which is a function of a double exponential law is known to hold if the random variables $X_i$ are mutually independent. When $M_n$ alone is considered Berman has shown that a double exponential law holds in the case of dependence provided either $r_n \log n \rightarrow 0$ or $\sum^\infty_{n=1} r_n^2 < \infty$. In the present work it is shown that the above conditions are also sufficient for the convergence of the joint law of $M_n$ and $S_n$. Weak convergence properties of the stochastic processes $M_{\lbrack nt\rbrack}$ and $S_{\lbrack nt \rbrack}$ with $0 < a \leqq t < \infty$ are also discussed.