Unusual limit theorems for the two tailed Pareto distribution

• Autorzy: Adler André , Matuła Przemysław

Identyfikatory

DOI

10.37190/0208-4147.41.2.4

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

almost sure convergence; order statistics; strong laws of large numbers; exact strong laws; fair games; weak law of large numbers; central limit theorems

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2021
• Tom: Vol. 41, Fasc. 2
• Strony: 267--281
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Adler André

• Department of Applied Mathematics Illinois Institute of Technology Chicago, IL 60616, USA

autor

Matuła Przemysław

• Institute of Mathematics Marie Curie-Skłodowska University Pl. M. Curie-Skłodowskiej 1 20-031 Lublin, Poland

Bibliografia

• [1] A. Adler, Laws of large numbers for two tailed Pareto random variables, Probab. Math. Statist. 28 (2008), 121-128.
• [2] A. Adler and A. Rosalsky, Some generalized central limit theorems for weighted sums with infinite variance, Calcutta Statist. Assoc. Bull. 38 (1989), 27-42.
• [3] Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 3rd ed., Springer, New York, 1997.
• [4] M. Klass and H. Teicher, Iterated logarithm laws for asymmetric random variables with or without finite mean, Ann. Probab. 5 (1977), 861-874.
• [5] P. Matuła, P. Kurasi ́nski, 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.
• [6] H. Xu, X. Li, W. Yang and F. Xu, Laws of large numbers with infinite mean, J. Math. Inequal. 13 (2019), 225-349.
• [7] 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.