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

Let ( X n ) be a stationary Gaussian sequence with mean 0 and variance 1. Let r n = E ( X 1 X n + 1 ) and M n = max { X k , 1 ≤ k ≤ n } . Suppose that some of the random variables of ( X n ) 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 , M n ) for the strongly dependent case where r n is convex with r n = o ( 1 ) and ( r n log n ) − 1 is monotone with ( r n log n ) − 1 = o ( 1 ) .