On the almost sure convergence of randomly indexed maximum of random variables
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 73 (2019) no. 2, pp. 91-104.

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We prove an almost sure random version of a maximum limit theorem, using logarithmic means for max_1≤ i≤ N_n X_i, where {X_n, n ≥ 1} is a sequence of identically distributed random variables and {N_n, n ≥ 1} is a sequence of positive integer random variables independent of {X_n, n ≥ 1}. Furthermore, we consider the almost sure random version of a limit theorem for kth order statistics.
Keywords: Almost sure central limit theorem, randomly indexed sums
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Krajka, Andrzej; Rychlik, Zdzisław; Wasiura-Maślany, Joanna. On the almost sure convergence of randomly indexed maximum of random variables. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 73 (2019) no. 2, pp. 91-104. http://geodesic.mathdoc.fr/item/AUM_2019_73_2_a1/

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