The uniform strong law of large numbers without any assumption on a family of sets
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
CHATTERJI S. D. (1972) Un principe de sous-suites dans la th´eorie des probabilit´es, S´eminaire de Probabilit´es VI Universit´e de Strasbourg (ed. N. Bourbaki), Springer, Berlin, pp. 72–89. Avalaible from: https://www.springer.com/gp/book/9783540057734
KOML´OS J. (1967) A generalization of a problem of Steinhaus, Acta Math. Hungar. 18 (1967) pp. 217–229. Avalaible from: https://link.springer.com/article/10.1007/BF02020976
CHATTERJI S. D. (1974) A principle of subsequences in probability theory: the central limit theorem, Adv. Math., 13, pp. 31–54. Avalaible from: https://pdf.sciencedirectassets.com/272585/1-s2.0-S0001870800X03969/1-s2.0-0001870874900644/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEKv
GAPOSHKIN V. F. (1966), Lacunary series and independent functions, Russian Mathematical Surveys, 21:6, 1–82. Avalaible from: http://mr.crossref.org/iPage?doi=10.1070
GUT A. (1985), On complete convergence in the law of large numbers for subsequences, Ann. Probab., vol. 13, no. 4, pp. 1286–1291. Avalaible from: https://projecteuclid.org/euclid.aop/1176992812
KLESOV O. I. (1990), On integrability of jSnk=nkj, in book New Trends in Probability and Statistics. Proceedings of the Bakuriani Colloquium in honour of Yu. V. Prokhorov, Bakuriani, Georgia, USSR, 24 February– 4 March 1990 (eds. V. V. Sazonov and T. Shervashidze), vol. 1, 1990, Mokslas and VSP, Vilnius, Lithuania, and Utrecht, the Netherlands, pp. 38–42.
BOGDANSKII, V. Yu., KLESOV, O. I., MOLCHANOV, I. (2019) Uniform strong law of large numbers, Methodol. Comput. Appl. Probab. Avalaible from: https://link.springer.com/article/10.1007/s11009-019-09711-x.
BOGDANSKII, V. Yu. and KLESOV, O. I. (2020) On a theorem of Pyke and Bass, Nauk. visnyk Uzhgorod Univ., №2, pp. 34–41. Avalaible from: https://doi.org/10.24144/2616-7700.2020.2(37).34-41
KLESOV O. I. and MOLCHANOV I. (2017) Moment conditions in strong laws of large numbers for multiple sums and random measures, Stat. Probab. Lett., vol. 131, pp. 56–63. Avalaible from: https://www.sciencedirect.com/science/article/abs/pii/S0167715217302675
KLESOV O. I. and MOLCHANOV I. (2019) Uniform strong law of large numbers, in book Modern Mathematics and Mechanics: Fundamentals, Problems and Challenges (editors V. A. Sadovnichiy and M. Z. Zgurovsky), Springer International Publishing AG, Cham (Switzerland), pp. 335–350. Avalaible from: https://link.springer.com/article/10.1007/s11009-019-09711-x
BASS R. F. and PYKE R. (1984) A strong law of large numbers for partial-sum processes indexed by sets Ann. Probab., vol. 12, issue 1, pp. 268–271.
KLESOV, O. I. Limit Theorems for Multi-Indexed Sums of Random Variables, Springer, Berlin, 2014, xviii+483 pp. Avalaible from: http://www.springer.com/mathematics/probability/book/978-3-662-44387-3
MARTIKAINEN, A. I. and PETROV, V. V. (1980) On a Theorem of Feller. Theor. Probab. Appl. vol. 25, issue 1, pp. 191–193. Avalaible from: https://epubs.siam.org/doi/10.1137/1125023
Feller, W. (1946) A limit theorem with infinite moments. Amer. J. Math., vol. 68, issue 2, pp. 257–262. Avalaible from: https://www.jstor.org/stable/2371837?seq=1
ETEMADI, N. (1981) An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheorie Verw. Geb. vol. 55, issue 1, pp. 119–122 Avalaible from: https://link.springer.com/article/10.1007/BF01013465
КLESOV O. I. (1983), Rate of convergence of series of random variables, Ukrainian Math. J., vol. 35, pp. 264–268. Avalaible from: https://link.springer.com/article/10.1007%2FBF01092173
КLESOV O. I. (1985), Rate of convergence of some random series, Theor. Probab. Math. Statist., vol. 30, pp. 91–101.
BULDYGIN V. V., KLESOV O. I., and Steinebach J. G. (2004) Properties of asymptotically quasi-inverse functions and their applications. I Theory Probab. Math. Statist. — vol. 70. — pp. 11–28. Avalaible from: http://probability.univ.kiev.ua/tims/issuesnew/70/PDF/4.pdf
BULDYGIN V. V., KLESOV O. I., and Steinebach J. G. (2008) On some properties of asymptotically quasi-inverse functions. Theory Probab. Math. Statist. — vol. 77. — pp. 15–30. Avalaible from: http://www.ams.org/journals/tpms/2005-70-00/S0094-9000-05-00627-7/S0094-9000-05-00627-7.pdf
How to Cite
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).