The asymptotic behaviour of maxima of complete and incomplete samples from stationary sequences

Let {Xn,n≥1}{Xn,n≥1} be a strictly stationary sequence of random variables and Mn=max{X1,X2,…,Xn}Mn=max{X1,X2,…,Xn}. Assume that some random variables X1,X2,…X1,X2,… can be observed and the sequence of random variables ε={εn,n≥1} indicate which X1,X2,…X1,X2,… are observed, thus Mn(ε)=max{Xj:εj=1,1≤j≤n}. In paper (Mladenovič and Piterbarg, 2006 [3]), the limiting behaviour (Mn,Mn(ε)) is investigated under the condition ∑j=1nεjn⟶Pp,as n→∞, for some real p∈(0,1)p∈(0,1). We generalize these results on the case, when for some random variable λλ∑j=1nεjn⟶Pλ,as n→∞.