Almost sure limit theorems of extremes of complete and incomplete samples of stationary sequences

Let (X*(n)) be an independent identically distributed random sequence. Let M*(n) and m*(n) denote, respectively, the maximum and minimum of {X*(1), ... , X*(n)}. Suppose that some of the random variables X*(1), X*(2), ... can be observed and let (M) over tilde*(n) and (m) over tilde*(n) denote, respectively, the maximum and minimum of the observed random variables from the set {X*(1), ... , X*(n)}. In this paper, we consider the asymptotic joint limiting distribution and the almost sure limit theorems related to the random vector ((M) over tilde*(n), (m) over tilde*(n), M*(n), m*(n)). The results are extended to weakly dependent stationary Gaussian sequences.