Limiting distributions and almost sure limit theorems for the normalized maxima of complete and incomplete samples from Gaussian sequence

Abstract

Let {X_k,k⩾1} be a stationary Gaussian sequence with partial maximum M_n=max {X_k,1⩽k⩽n} and sample mean X̅_n=∑_k=1ⁿX_k/n. Suppose that some of the random variables X₁,X₂,… can be observed and the others not. Denote by M̃_n the maximum of the observed random variables from the set {X₁,X₂,…,X_n}. Under some mild conditions, we prove the joint limiting distribution and the almost sure limit theorem for (M̃_n−X̅_n,M_n−X̅_n).