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


Aksomaitis, A., Transfer theorems in a max-scheme, Litovsk. Mat. Sb. 29 (2) (1989), 207–211 (Russian).

Barndorff-Nielsen, O., On the limit distribution of the maximum of a random number of independent random variables, Acta. Math. Acad. Sci. Hungar. 11 (1964), 399–403.

Berkes, I., Csaki, E., A universal result in almost sure central limit theory, Stoch. Proc. Appl. 94 (2001), 105–134.

Brosamler, G. A., An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561–574.

Berman, S. M., Limiting distribution of the maximum in the sequence of dependent random variables, Ann. Math. Statist. 33 (1962), 894–908.

Cheng, S., Peng, L., Qi, Y., Almost sure convergence in extreme value theory, Math. Nachr. 190 (1998), 43–50.

Fahrner, I., Almost Sure Versions of Weak Limit Theorems, Shaker Verlag, Aachen, 2000.

Fahrner, I., Stadmuller, U., On almost sure max-limit theorems, Statist. Probab. Lett. 37 (1998), 229–236.

Galambos, J., The Asymptotic Theory of Extreme Order Statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York–Chichester–Brisbane, 1978.

Hormann, S., An extension of almost sure central limit theory, Statist. Probab. Lett. 76 (2006), 191–202.

Jakubowski, A., Asymptotic Independent Representations for Sums and Order Statistics of Stationary Sequences, NCU Press publications, Toruń, 1991.

Krajka, A., Wasiura, J., On the almost sure central limit theorem for randomly indexed sums, Math. Nachr. 282 (4) (2009), 569–580.

Lacey, M. T., Philipp, W., A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), 201–205.

Leadbetter, M. R., Lindgren, G., Rootzen, H., Extremes and Related Properties of Random Sequences and Processes, Springer, Berlin, 1983.

O’Brien, G. L., The maximum term of uniformly mixing stationary process, Z. Wahr. verw. Gebiete 30 (1974), 57–63.

Robbins, H., The asymptotic distribution of the sums of a random number of random variables, Bull. Amer. Math. Soc. 54 (1948), 1151–1161.

Resnick, S. I., Extreme Values. Regular Variation and Point Processes, Springer, 1987.

Stadtmuller, U., Almost sure versions of distributional limit theorems for certain order statistics, Statist. Probab. Lett. 58 (2002), 413–426.

Schatte, P., On strong version of the central limit theorem, Math. Nachr. 137 (1988), 249–256.




DOI: http://dx.doi.org/10.17951/a.2019.73.2.91-104
Date of publication: 2020-01-16 07:29:34
Date of submission: 2020-01-08 13:15:22


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