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

## 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

## References

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*Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics*, (3), 39–48. https://doi.org/10.17721/1812-5409.2020/3.4

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