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

Let { X n , n ≥ 1 } be a strictly stationary sequence of random variables and M n = max { X 1 , X 2 , … , X n } . Assume that some random variables X 1 , X 2 , … can be observed and the sequence of random variables ε = { ε n , n ≥ 1 } indicate which X 1 , X 2 , … are observed, thus M n ( ε ) = max { X j : ε j = 1 , 1 ≤ j ≤ n } . In paper (Mladenovič and Piterbarg, 2006 [3]), the limiting behaviour ( M n , M n ( ε ) ) is investigated under the condition ∑ j = 1 n ε j n ⟶ P p , as  n → ∞ , for some real p ∈ ( 0 , 1 ) . We generalize these results on the case, when for some random variable λ ∑ j = 1 n ε j n ⟶ P λ , as  n → ∞ .