The uniform strong law of large numbers without any assumption on a family of sets

Authors

  • V. Yu. Bogdanskii National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, 03056 Kyiv https://orcid.org/0000-0001-5334-8471
  • O. I. Klesov National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, 03056 Kyiv https://orcid.org/0000-0002-0365-7716

DOI:

https://doi.org/10.17721/1812-5409.2020/3.4

Abstract

We study the sums of identically distributed random variables whose indices belong to certain sets of a given family A in R^d, d >= 1. We prove that sums over scaling sets S(kA) possess a kind of the uniform in A strong law of large numbers without any assumption on the class A in the case of pairwise independent random variables with finite mean. The well known theorem due to R. Bass and R. Pyke is a counterpart of our result proved under a certain extra metric assumption on the boundaries of the sets of A and with an additional assumption that the underlying random variables are mutually independent. These assumptions allow to obtain a slightly better result than in our case. As shown in the paper, the approach proposed here is optimal for a wide class of other normalization sequences satisfying the Martikainen–Petrov condition and other families A. In a number of examples we discuss the necessity of the Bass–Pyke conditions. We also provide a relationship between the uniform strong law of large numbers and the one for subsequences.

Key words: sums of random variables, uniform in a family of sets limit results, strong law of large numbers.

Pages of the article in the issue: 39 - 48

Language of the article: Ukrainian

Author Biographies

V. Yu. Bogdanskii, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, 03056 Kyiv

Аспірант кафедри математичного аналізу та теорії ймовірностей

O. I. Klesov, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, 03056 Kyiv

Завідувач кафедри математичного аналізу та теорії ймовірностей, доктор фізико-математичних наук, професор

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No 3 (2020)

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How to Cite

Bogdanskii, V. Y., & Klesov, O. I. (2020). The uniform strong law of large numbers without any assumption on a family of sets. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, (3), 39–48. https://doi.org/10.17721/1812-5409.2020/3.4

Issue

No. 3 (2020)

Section

Algebra, Geometry and Probability Theory

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