On exact strong laws of large numbers under general dependence conditions

• Autorzy: Adler A. , Matuła P.

Identyfikatory

DOI

10.19195/0208-4147.38.1.6

• Języki publikacji: EN

Abstrakty

EN

We study the almost sure convergence of weighted sums of dependent random variables to a positive and finite constant, in the case when the random variables have either mean zero or no mean at all. These are not typical strong laws and they are called exact strong laws of large numbers. We do not assume any particular type of dependence and furthermore consider sequences which are not necessarily identically distributed. The obtained results may be applied to sequences of negatively associated random variables.

Słowa kluczowe

EN

strong law of large numbers; almost sure convergence; slowly varying functions; negative association

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2018
• Tom: Vol. 38, Fasc. 1
• Strony: 103--121
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Adler A.

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

autor

Matuła P.

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

Bibliografia

• [1] A. Adler, Exact strong laws, Bull. Inst. Math. Acad. Sin. 28 (2000), pp. 141-166.
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• [3] T. K. Chandra and S. Ghosal, The strong law of large numbers for weighted averages under dependence assumptions, J. Theoret. Probab. 9 (1996), pp. 797-809.
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• [6] I. Fazekas, P. Matuła, and M. Ziemba, A note on the weighted strong law of large numbers under general conditions, Publ. Math. Debrecen 9 (2017), pp. 373-386.
• [7] D. H. Hong and J. M. Park, Exact sequences for sums of pairwise i.i.d. random variables, Bull. Korean Math. Soc. 30 (1993), pp. 167-170.
• [8] K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), pp. 286-295.
• [9] M. Klass and H. Teicher, Iterated logarithm laws for asymmetric random variables barely with or without finite mean, Ann. Probab. 5 (1977), pp. 861-874.
• [10] E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist. 37 (1966), pp. 1137-1153.
• [11] R. A. Maller, Relative stability and the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 43 (1978), pp. 141-148.
• [12] P. Matuła, A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab. Lett. 15 (1992), pp. 209-213.
• [13] V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables, Clarendon Press, Oxford 1995.
• [14] L. X. Zhang and X. Y. Wang, Convergence rates in the strong laws of asymptotically negatively associated random fields, Appl. Math. J. Chinese Univ. 14 (1999), pp. 406-416.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).