On the almost sure convergence of randomly indexed maximum of random variables

Andrzej Krajka, Zdzisław Rychlik, Joanna Wasiura-Maślany

Abstract


We prove an almost sure random version of a maximum limit theorem, using logarithmic means for \(\max_{1\leq i\leq N_n} X_i\), where \(\{X_n, n \geq 1\}\) is a sequence of identically distributed random variables and \(\{N_n, n \geq 1\}\) is a sequence of positive integer random variables independent of \(\{X_n, n \geq 1\}\). Furthermore, we consider the almost sure random version of a limit theorem for \(k\)th order statistics.

Keywords


Almost sure central limit theorem; randomly indexed sums

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References


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DOI: http://dx.doi.org/10.17951/a.2019.73.2.91-104
Data publikacji: 2020-01-16 07:29:34
Data złożenia artykułu: 2020-01-08 13:15:22


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