Maxima of Partial Samples in Gaussian Sequences

Abstract

Let $\{X_n, n \geqq 0\}$ be a Gaussian sequence with $EX_i \equiv 0; V(X_i) \equiv 1$ and $EX_iX_j = r_{ij}$. Define $M_n = \max_{0\leqq i \leqq n}X_i$ and $m_{n,k} = \max (X_i; i \in G_n)$ where $G_n = (t_1, \cdots, t_{n'})$ is a subset of $(0, 1, 2, \cdots, n)$ and $n' = \lbrack n/k\rbrack$ for some integer $k \geqq 1, \lbrack x\rbrack$ being the integral part of $x$. We show that $P\{c_n(M_n - m_{n, k}) \leqq x\} \rightarrow (1 + (k - 1)e^{-x})^{-1}$ as $n \rightarrow \infty$ for all $x \geqq 0$ where $c_n = (2 \log n)^\frac{1}{2}$, if the sequence is "moderately dependent," namely if \begin{equation*}\tag{1}(i) \sup_{ij} |r_{ij}| < \delta < 1\end{equation*} $$(ii) |r_{ij}| \leqq \rho(i - j) \text{for} |i - j| > N_0 \text{such that} \rho_n\log n = o(1).$$ Somewhat surprisingly the same result holds even though the sequence is "strongly dependent," namely if \begin{equation*}\tag{2}(i) r_{ij} = r(i - j); r_n \text{convex for} n \geqq 0 \text{and} r_n = o(1)\end{equation*} $$(ii) (r(n) \log n)^{-1} \text{is monotone and} o(1).$$