Asymptotic distributions of maxima of complete and incomplete samples from strongly dependent stationary Gaussian sequences

Let (Xn) be a stationary Gaussian sequence with mean 0 and variance 1. Let rn=E(X1Xn+1) and Mn=max{Xk,1≤k≤n}. Suppose that some of the random variables of (Xn) can be observed and let M˜n denote the partial maximum of the observed variables. In this note, we study the limiting distribution of random vector (M˜n,Mn) for the strongly dependent case where rn is convex with rn=o(1) and (rnlogn)−1 is monotone with (rnlogn)−1=o(1).