An iterated logarithm law for the maximum in a stationary gaussian sequence

Abstract

Let {X n , n=1, 2, ⋯} be the successive terms of a discrete coordinate stationary Gaussian stochastic process. Assume, without loss of generality, that EX n =0 and r o =EX n 2 =1 for all n. Let r n ≡EX k X k+n be the covariance function. If either there exists an α>0 such that $$\mathop {\lim }\limits_{n \to \infty } n^\alpha r_n = 0, or \sum\limits_{n = - \infty }^\infty {r_n^2 < \infty ,}$$ then $$P\{ \mathop {\lim inf}\limits_{n \to \infty } (2\log n)^{\tfrac{1}{2}} (Z_n - (2\log n)^{\tfrac{1}{2}} )/\log log n = - \tfrac{1}{2}, \mathop {limsup}\limits_{n \to \infty } (2\log n)^{\tfrac{1}{2}} (Z_n - (2\log n)^{\tfrac{1}{2}} )/\log log n = - \tfrac{1}{2}\} = 1,$$ where $$Z_n \equiv \mathop {Sup}\limits_{1 \leqq k \leqq n} X_k .$$ It is not sufficient that $$\mathop {lim}\limits_{n \to \infty } r_n = 0.$$