The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences

Abstract

Suppose that X 1 , X 2 , … is a standardized stationary Gaussian sequence. Let: M n ≔ max ( X 1 , … , X n ) , S n ≔ ∑ i = 1 n X i , σ n ≔ Var ( S n ) , and a n > 0 , b n denote suitable normalizing constants. Our goal is to prove the almost sure central limit theorem for the sequence { a n ( M n - b n ) , S n / σ n } , under certain additional assumptions on the covariance function r ( t ) ≔ Cov ( X 1 , X 1 + t ) .