Almost Sure Limit Points of Maxima of Stationary Gaussian Sequences

Abstract

Let $\{X_n, n \geqslant 1\}$ be a discrete-parameter stationary Gaussian process with $E(X_i) = 0, E(X^2_i) = 1$ for all $i$ and $E(X_iX_{i+n}) = r(n)$. Let $M_n =$ maximum$(X_1, X_2 \cdots X_n)$. Under the condition that either $(\log n)^{1+\gamma}r(n) = O(1)$ as $n \rightarrow \infty$, for some $\gamma > 0$ or $\sum^\infty_{j=1}r^2(j) < \infty$, the set of all almost sure limit points of the vector sequence $$\big\{\frac{M_{1,n} - b_n}{a_n}, \frac{M_{2,n} - b_n}{a_n}, \cdots \frac{M_{p,n} - b_n}{a_n}\big\}$$ is obtained, where $(M_{j,n}), j = 1,2 \cdots p$ are independent copies of $(M_n); a_n = (\log\log n)(2 \log n)^{-\frac{1}{2}}$ and $b_n = (2 \log n)^{\frac{1}{2}}$.