A law of iterated logarithm for stationary Gaussian processes

Abstract:

In this article the following results are established. Theorem A. Let $\{ X(t):0 \leqslant t < \infty \}$ be a stationary Gaussian process with continuous sample functions and $E[X(t)] \equiv 0$. Suppose that the covariance function $r(t)$ satisfies the following conditions. (a) $r(t) = 1 - |t{|^\alpha }H(t) + o(|t{|^\alpha }H(t))$ as $t \to 0$, where $0 < \alpha \leqslant 2$ and $H$ varies slowly at zero, and (b) $r(t) = O(1/\log t)$ as $t \to \infty$ Then for any nondecreasing positive function $\phi (t)$ defined on $[a,\infty )$ with $\phi (\infty ) = \infty ,P[X(t) > \phi (t)$ i.o. for some sequence ${t_n} \to \infty ] = 0or1$ according as the integral $I(\phi ) = \int _a^\infty {g(\phi (t))\phi {{(t)}^{ - 1}}\exp ( - {\phi ^2}(t)/2)dt}$ is finite or infinite, where $g(x) = 1/_\sigma ^{ \sim - 1}(1/x)$ is a regularly varying function with exponent $2/\alpha$ and $_\sigma ^{ \sim 2}(t) = 2|t{|^\alpha }H(t)$.

Theorem C. Let $\{ {X_n}:n \geqslant 1\}$ be a stationary Gaussian sequence with zero mean and unit variance. Suppose that its covariance function satisfies, for some $\gamma > 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty$. Let $\{ \phi (n):n \geqslant 1\}$ be a nondecreasing sequence of positive numbers with ${\lim _{n \to \infty }}\phi (n) = \infty$; suppose that $\Sigma (1/\phi (n))\exp ( - {\phi ^2}(n)/2) = \infty$. Then \[ \lim _{n \to \infty } \sum _{1 \leq k \leq n} I_k / \sum _{1 \leq k \leq n} E[I_k] = 1\quad \mathrm {a.s.}, \] where $I_k$ denotes the indicator function of the event $\{ {X_k} > \phi (k)\}$.