The uniform strong law of large numbers without any assumption on a family of sets
Тип публікації :
Стаття
Дата випуску :
2020
Автор(и) :
Bogdanskii, V. Yu.
Klesov, O. I.
Мова основного тексту :
Англійська
eKNUTSHIR URL :
Випуск :
3
ISSN :
1812-5409
Початкова сторінка :
39
Кінцева сторінка :
48
Цитування :
[APA 7] 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
[ДСТУ] Bogdanskii V. Y., Klesov O. I. 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. 2020. no. 3. P. 39—48. DOI: 10.17721/1812-5409.2020/3.4 (date of access: 25.07.2026).
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 - 48Language of the article: Ukrainian
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