DOI: https://doi.org/10.20535/1810-0546.2015.4.48791

### The Convergence Rate in Precise Asymptotics for Series of Large Deviations

Юлія Олександрівна Грегуль

#### Abstract

Background. In the paper the sequence of independent identically distributed random variables is considered. We are interested in conditions for the convergence of series for different values of parameters , and any . Such series appears while studying complete convergence as well as investigating various problems on large deviations in limit theorems of probability theory. The new approach to study such series is proposed by C. Heyde, who showed that if  then Further, this result was extended by O. Klesov, and finally generalized by A. Gut, J. Steinebach and J. Hi for the series with , and .

Objective. We consider the series for one-side deviations. The main purpose of the paper is to study precise asymptotics of function while Methods. The methods used to prove the main result is as follows: first we find the asymptotics for the partial case, i.e. we assume that random variables are Gaussian random variables, further we extend obtained result to the general case by means of estimations of rate of convergence in the Central limit theorem.

Results. In the paper the precise asymptotics of series while for is obtained. Strict assumptions imposed on parameter are connected with the application of Nagaev inequality on the rate of convergence in the Central limit theorem.

Conclusions. The asymptotics ought to be considered for other values of as well. This requires though new techniques since the use of Nagaev inequality leads to some restrictions upon .  Since series converges (under some moment conditions) for any then further investigations may concern the behavior of this series for and #### Keywords

Precise asymptotics; Convergence rate; Series of large deviations

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#### References

P.L. Hsu and H. Robbins, “Complete convergence and the law of large numbers”, Proc. Nat. Acad. Sci.USA, vol. 33, pp. 25–31, 1947.

P. Erdös, “On a theorem of Hsu and Robbins”, Ann. Math. Statist., vol. 20, pp. 287–291, 1949.

F. Spitzer, “A combinatorial lemma and its application to probability theory”, Trans. Amer. Math. Soc., vol. 82, pp. 323–339, 1956.

M. Katz, “The probability in the tail of distribution”, Ann. Math. Statist., vol. 34, pp. 312–318, 1963.

L.E. Baum, M. Katz, “Convergence rates in the law of large numbers”, Trans. Amer. Math. Soc., vol. 120, pp. 108–123, 1965.

C.C. Heyde, “A supplement to the strong law of large numbers”, J. Appl. Probab., vol. 12, pp. 903–907, 1975.

O.I. Klesov, “On the convergence rate in a theorem of Heyde”, Theory Probab. Math. Stat., vol. 49, pp. 83–87, 1994.

A. Gut, J. Steinebach, “Precise asymptotics – a general approach”, Acta Math. Hung., vol. 138, pp. 365–385, 2013.

J. He, “A note to the convergence rates in precise asymptotics”, J. Inequalities Appl., vol. 378, 2013 [Online]. Avaliable: http://www.journalofinequalitiesandapplications.com/content/2013/1/378

S.V. Nagaev, “Some limit theorems for large deviations”, Teor. Veroyatnost. I Primenen., vol. 10, pp. 231–254, 1965.

#### GOST Style Citations

1. Hsu P.L., Robbins H. Complete convergence and the law of  large numbers // Proc. Nat. Acad. Sci. USA. – 1947. – 33. – P. 25–31.

2. Erdös P. On a theorem of Hsu and Robbins // Ann. Math. Statist. – 1949. – 20. – P. 287–291.

3. Spitzer F. A combinatorial lemma and its application to probability theory// Trans. Amer. Math. Soc. – 1956. – 82. – P. 323–339.

4. Katz M. The probability in the tail of distribution // Ann. Math. Statist. – 1963. – 34. – P. 312–318.

5. Baum L.E., Katz M. Convergence rates in the law of large numbers // Trans. Amer. Math. Soc. – 1965. – 120. – P. 108–123.

6. Heyde C.C. A supplement to the strong law of large numbers // J. Appl. Probab. – 1975. – 12. – P. 903–907.

7. Klesov O.I. On the convergence rate in a theorem of Heyde // Theory Probab. Math. Stat. – 1994. – 49. – P. 83–87.

8. Gut A., Steinebach J. Precise asymptotics – a general approach // Acta Math. Hung. – 2013. – 138. – P. 365–385.

9. He J. A note to the convergence rates in precise asymptotics // J. Inequalities Appl. – 2013. – 378. – Режим доступу: http://www.journalofinequalitiesandapplications.com/content/2013/1/378.

10. Nagaev S.V. Some limit theorems for large deviations // Teor. Veroyatnost. I Primenen. – 1965. – 10. – P. 231–254.