On extreme values in incomplete samples from stationary sequences

Let $(X_n)$ be a strictly stationary random sequence with the marginal distribution functions $F(x)=P\{X_1\le x\}$. Suppose that some of random variables $X_1,X_2,X_3,\dots$ can be observed, and let $\varepsilon_k$ be the indicator of the event that random variables $X_k$ is observed and $S_n=\varepsilon_1+\dots+\varepsilon_n$. Let us denote $M_n=\max\{X_1,\dots,X_n\}$, $\tilde M_n=\max\{X_j:1\le j\le n,\ \varepsilon_j=1\}$.
Suppose that $F$ belongs to the maximum domain of attraction of some of extreme value distributions. The limiting distributions of the random vector $(\tilde M_n,M_n)$ and “asymptotic independency” of $\tilde M_n$ and $M_n$ are obtained under some condition of weak dependency of random variables from sequence $(X_n)$, which is more restrictive than Leadbetter's $D(u_n)$ condition, and some conditions on the sequence $(\varepsilon_n)$.
Some results concerning estimation of the exponent of regular variation using a sample with the missing observations will also be presented.