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

Suppose that X1,X2,…X1,X2,… is a standardized stationary Gaussian sequence. Let: Mn≔max(X1,…,Xn)Mn≔max(X1,…,Xn), Sn≔∑i=1nXi, σn≔Var(Sn), and an>0an>0, bnbn denote suitable normalizing constants. Our goal is to prove the almost sure central limit theorem for the sequence {an(Mn-bn),Sn/σn}{an(Mn-bn),Sn/σn}, under certain additional assumptions on the covariance function r(t)≔Cov(X1,X1+t)r(t)≔Cov(X1,X1+t).