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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 41, Fasc. 2,
pages 267 - 281
DOI: 10.37190/0208-4147.41.2.4
Published online 11.8.2021
 
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Unusual limit theorems for the two-tailed Pareto distribution

A. Adler
P. Matuła

Abstract:

We examine order statistics from a two-sided Pareto distribution. It turns out that the smallest two order statistics and the largest two order statistics have very unusual limits. We obtain strong and weak exact laws for the smallest and the largest order statistics. For such statistics we also study the generalized law of the iterated logarithm. For the second smallest and second largest order statistics we prove the central limit theorem even though their second moment is infinite.

2010 AMS Mathematics Subject Classification: Primary 60F15; Secondary 60F05.

Keywords and phrases: almost sure convergence, order statistics, strong laws of large numbers, exact strong laws, fair games, weak law of large numbers, central limit theorems.

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Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 3rd ed., Springer, New York, 1997.

M. Klass and H. Teicher, Iterated logarithm laws for asymmetric random variables with or without finite mean, Ann. Probab. 5 (1977), 861-874.

P. Matuła, P. Kurasiński, H. Naderi and A. Adler, A note on exact laws of large numbers for asymmetric Pareto-type distributions with applications to ratios of random variables, Bull. Inst. Math. Acad. Sinica (N.S.) 15 (2020), 177-186.

H. Xu, X. Li, W. Yang and F. Xu, Laws of large numbers with infinite mean, J. Math. Inequal. 13 (2019), 225-349.

W. Yang, L. Yang, D. Wei and S. Hu, The laws of large numbers for Pareto-type random variables with infinite means, Comm. Statist. 48 (2019), 3044-3054.

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