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On the asymptotically uniform distribution modulo 1 of extreme order statistics. (English) Zbl 0826.62013

Summary: Let \((X_m )^\infty_1\) be a sequence of independent and identically distributed random variables. We give sufficient conditions for the fractional part of \(\max (X_1, \dots, X_n)\) to converge in distribution, as \(n\to \infty\), to a random variable with a uniform distribution on \([0,1)\).

MSC:

62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
60F05 Central limit and other weak theorems
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References:

[1] J. J. A. M. Brands(1991 ), An asymptotic problem in extremal processes, RANA 91-10, Eindhoven University of Technology, Eindhoven, The Netherlands.
[2] J. J. A. M. Brands, F. W. Steutel, and R. J. G. Wilms(1992 ), On the number of maxima in a discrete sample, Memorandum COSOR 92-16, Eindhoven University of Technology, Eindhoven, The Netherlands (to appear in Statistics and Probability Letters 20).
[3] DOI: 10.1214/aoms/1177697624 · Zbl 0176.47502
[4] Jagers A. A., Statistica Neerlandica 44 pp 180– (1990)
[5] Kuipers L., Uniform distribution of sequences (1974) · Zbl 0281.10001
[6] DOI: 10.2307/2319349 · Zbl 0349.60014
[7] Resnick S. I., Extreme values, regular variation and point processes (1987) · Zbl 0633.60001
[8] Ripley B. D., Stochastic simulation (1987) · Zbl 0613.65006
[9] DOI: 10.1002/mana.19831100118 · Zbl 0523.60016
[10] Zygmund A., Trigonometric series 11 (1968) · Zbl 0157.38204
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