ALMOST SURE CENTRAL LIMIT THEOREMS OF THE PARTIAL SUMS AND MAXIMA FROM COMPLETE AND INCOMPLETE SAMPLES OF STATIONARY SEQUENCES

Let (X-n) denote an independent and identically distributed random sequence. Let S-n = Sigma(n)(k=1) X-k and M-n = max{X-1, ... , X-n} be its partial sum and maximum. Suppose that some of the random variables of X-1, X-2, ... can be observed and denote by (M) over tilde (n) the maximum of observed random variables from the set {X-1, ... , X-n}. In this paper, we consider the joint limiting distribution of ((M) over tilde (n), M-n, S-n) and the almost sure central limit theorems related to the random vector ((M) over tilde (n), M-n, S-n). Furthermore, we extend related results to weakly dependent stationary Gaussian sequences.