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

Abstract

Let $(X_{n}^{\ast})$ be an independent identically distributed random sequence. Let $M_{n}^{\ast}$ and $m_{n}^{\ast}$ denote, respectively, the maximum and minimum of $\{X_{1}^{\ast},\cdots,X_{n}^{\ast}\}$. Suppose that some of the random variables $X_1^{\ast},X_2^{\ast},\cdots$ can be observed and let $\widetilde{M}_n^{\ast}$ and $\widetilde{m}_n^{\ast}$ denote, respectively, the maximum and minimum of the observed random variables from the set $\{X_1^{\ast},\cdots,X_n^{\ast}\}$. In this paper, we consider the asymptotic joint limiting distribution and the almost sure limit theorems related to the random vector $(\widetilde{M}_n^{\ast}, \widetilde{m}_n^{\ast}, M_n^{\ast}, m_n^{\ast})$. The results are extended to weakly dependent stationary Gaussian sequences.