On probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition
DOI:
https://doi.org/10.21638/spbu01.2023.308Abstract
We derive new results on asymptotic behaviour for probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition. We find zones in which these probabilities are equivalent to the tail of the standard normal law. The author earlier obtained such results under Bernstein’s condition. The truncations method is applied in proofs of results.
Keywords:
probabilities of large deviations, combinatorial central limit theorem, combinatorial sums
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Литература
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References
1. Wald A., Wolfowitz J. Statistical tests based on permutations of observations. Ann. Math. Statist. 15, 358-372 (1944).
2. Noether G. E. On a theorem by Wald and Wolfowitz. Ann. Math. Statist. 20, 455-458 (1949).
3. Hoeffding W. A combinatorial central limit theorem. Ann. Math. Statist. 22, 558-566 (1951).
4. Motoo M. On Hoeffding’s combinatorial central limit theorem. Ann. Inst. Statist. Math. 8, 145-154 (1957).
5. Kolchin V. F., Chistyakov V. P. Оn a combinatorial limit theorem. Teoriia veroiatnostei i ee primenenie 18 (4), 767-777 (1973). (In Russian) [Engl. trans.: Theory of Probability and its Applications 18, (4), 728-7391974. https://doi.org/10.1137/1118093].
6. Bolthausen E. An estimate of the remainder in a combinatorial central limit theorem. Zeitschrift f¨ur Wahrsch. und Verwandte Geb. 66, 379-386 (1984).
7. von Bahr B. Remainder term estimate in a combinatorial central limit theorem. Zeitschrift f¨ur Wahrsch. und Verwandte Geb. 35, 131-139 (1976).
8. Ho S. T., Chen L. H. Y. An Lp bounds for the remainder in a combinatorial central limit theorem. Ann. Probab. 6, 231-249 (1978).
9. Goldstein L. Berry-Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42, 661-683 (2005).
10. Neammanee K., Suntornchost J. A uniform bound on a combinatorial central limit theorem. Stoch. Anal. Appl. 3, 559-578 (2005).
11. Neammanee K., Rattanawong P. A constant on a uniform bound of a combinatorial central limit theorem. J. Math. Research 1, 91-103 (2009).
12. Chen L. H. Y., Goldstein L., Shao Q. M. Normal approximation by Stein’s method. Springer (2011).
13. Chen L. H. Y., Fang X. Оn the error bound in a combinatorial central limit theorem. Bernoulli 21 (1), 335-359 (2015).
14. Frolov A. N. Esseen type bounds of the remainder in a combinatorial CLT. J. Statist. Planning and Inference 149, 90-97 (2014).
15. Frolov A. N. Bounds of the remainder in a combinatorial central limit theorem. Statist. Probab. Letters 105, 37-46 (2015).
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18. Frolov A. N. On large deviations for combinatorial sums. J. Statist. Planning and Inference 217, 24-32 (2022).
19. Linnik Iu. V. Limit theorems for sums of independent random variables. I, II, III. Teoriia veroiatnostei i ee primenenie 6 (2), 145-163 (1961); 6 (4), 377-391 (1961); 7 (2), 121-134 (1962). (In Russian)
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Published
2023-09-23
How to Cite
Frolov А. N. (2023). On probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 10(3), 545–553. https://doi.org/10.21638/spbu01.2023.308
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
