Limiting Behavior of Maxima in Stationary Gaussian Sequences

Abstract

Let $\{X_n, n \geqq 1\}$ be a real-valued, stationary Gaussian sequence with mean zero and variance one. Let $M_n = \max_{1\leqq i\leqq n} X_i, r_n = E(X_{n+1}X_1); c_n = (2 \ln n)^{\frac{1}{2}}$ and $b_n = c_n - \frac{1}{2}\lbrack\ln (4\pi \ln n)\rbrack/c_n$. Define $U_n = 2c_n(M_n - c_n)/\ln\ln n$ and $V_n = c_n(M_n - b_n)$. If $r_n = O(1/\ln n)$ as $n \rightarrow \infty$ then (i) $p(\lim \inf_{n\rightarrow\infty} U_n = -1) = p(\lim \sup_{n\rightarrow\infty} U_n = 1) = 1$, and (ii) $E\{\exp(tV_n)\} \rightarrow E\{\exp (tX)\}$ as $n \rightarrow \infty$ for all $t$ sufficiently small where $X$ is a random variable with distribution function $e^{-e^{-x}}; -\infty < x < \infty$.