The asymptotic behaviour of maxima of complete and incomplete samples from stationary sequences
Assume that, from a strictly stationary sequence \(\{X_n,\;n\geq 1\}\), only a random subsequence, given by an independent sequence \(\varepsilon=\{\varepsilon_n,\;n\geq 1\}\) of indicators, can be observed. Let \(M_n=\max\{X_1,\cdots,X_n\}\) and \(M_n(\varepsilon)=\max \{X_j:\varepsilon_j=1, 1\leq j\leq n\}\) (i.e., the maxima of the observed random variables \(X_j\), where \(1\leq j\leq n\)). Consider a random variable \(\lambda\) taking values in \([0,1]\). The paper investigates the asymptotic behavior of \((M_n,M_n(\varepsilon))\) under the condition that \(\sum_{j=1}^n\varepsilon_j/n\overset{\text P}{\rightarrow} \lambda,\) as \(n\to\infty\). Thus, the author extends the results of \textit{P. Mladenovič} and \textit{V. Piterbarg} [Stochastic Processes Appl. 116, No. 12, 1977--1991 (2006; Zbl 1118.60047)]. Four examples enrich this presentation.
scientific article; zbMATH DE number 5930907