On probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition

Authors

  • Аndrei N. Frolov St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2023.308

Abstract

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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References

Литература

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. Колчин В. Ф., Чистяков В. П. Об однойкомбинаторнойпредельнойтеореме. Теория вероятностей и ее применение 18 (4), 767-777 (1973).

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).

16. Фролов А. Н. О вероятностях умеренных уклоненийкомбинаторных сумм. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 2 (60), вып. 1, 60- 67 (2015).

17. Frolov A. N. On Esseen type inequalities for combinatorial random sums. Communications in Statistics-Theory and Methods 46 (12), 5932-5940 (2017).

18. Frolov A. N. On large deviations for combinatorial sums. J. Statist. Planning and Inference 217, 24-32 (2022).

19. Линник Ю. В. Предельные теоремы для сумм независимых случайных величин I, II, III. Теория вероятностей и ее применение 6 (2), 145-163 (1961); 6 (4), 377-391 (1961); 7 (2), 121-134 (1962).

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).

16. Frolov A. N. On the probabilities of moderate deviations for combinatorial sums. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 2 (60), iss. 1, 60-67 (2015). (In Russian) [Engl. trans.: Vestnik St. Petersburg University. Mathematics 48, iss. 1, 23-28 (2015). https://doi.org/10.3103/S1063454115010045]

17. Frolov A. N. On Esseen type inequalities for combinatorial random sums. Communications in Statistics-Theory and Methods 46 (12), 5932-5940 (2017).

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)

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

Issue

Section

Mathematics