The strong law of large numbers for dependent random variables. (English) Zbl 1082.60023

Kuczmaszewska, Anna

Stat. Probab. Lett. 73, No. 3, 305-314 (2005).

This paper establishes two results (sufficient conditions) on the strong law of large numbers under negative association (Theorem 2) and under \(\rho\)-mixing (Theorem 5), respectively. Some other results on sums of negatively associated random variables were obtained by P. Matuła [Stat. Probab. Lett. 15, No. 3, 209–213 (1992; Zbl 0925.60024)]. The definition of negative association due to J. D. Esary, F. Proschan and D. W. Walkup [Ann. Math. Stat. 38, 1466–1474 (1967; Zbl 0183.21502)] and K. Joag-Dey and F. Proschan [Ann. Stat. 11, 286–295 (1983; Zbl 0508.62041)] and the definition of \(\rho\)-mixing based on the Kolmogorov-Rozanov mixing coefficient \(\rho(n)\) are given.
Theorem 5 generalizes the results of Q.-M. Shao [Ann. Probab. 23, No. 2, 948–965 (1995; Zbl 0831.60028)] and I. Fazekas and O. Klesov [Theory Probab. Appl. 45, No. 3, 436–449 (2000) and Teor. Veroyatn. Primen. 45, No. 3, 568–583(2000; Zbl 0991.60021)]. The results are obtained by using the maximal inequality of J. Hájek andA. Rényi [Acta Math. Acad. Sci. Hung. 6, 281–283 (1956; Zbl 0067.10701)].

Reviewer: Ludwig Paditz (Dresden)

MSC:

60F15	Strong limit theorems
60G50	Sums of independent random variables; random walks

Keywords:

negatively associated random variables; \(\rho\)-mixing sequence; Háyek-Rényi-type maximal inequality

Citations:

Zbl 0925.60024; Zbl 0183.21502; Zbl 0508.62041; Zbl 0831.60028; Zbl 0991.60021; Zbl 0067.10701

References:

[1]	Chung, K. L., Note on some strong laws of large numbers, Amer. J. Math., 69, 189-192 (1947) · Zbl 0034.07103
[2]	Esary, J.; Proschan, F.; Walkup, D., Association of random variables with applications, Ann. Math. Statist., 38, 1466-1474 (1967) · Zbl 0183.21502
[3]	Fazekas, I.; Klesov, O., A general approach to the strong laws of large numbers, Teor. Verojatnost. i Primenen., 45, 569-583 (2000) · Zbl 0991.60021
[4]	Háyek, J.; Rényi, A., Generalization of an inequality of Kolmogorov, Acta. Math. Acad. Sci. Hungar., 6, 281-283 (1955) · Zbl 0067.10701
[5]	Joag-Dev, K.; Proschan, F., Negative association of random variables with applications, Ann. Statist., 11, 286-295 (1983) · Zbl 0508.62041
[6]	Matuła, P., A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab. Lett., 15, 209-213 (1992) · Zbl 0925.60024
[7]	Révész, P., The Laws of Large Numbers (1968), Academic Press: Academic Press New York · Zbl 0203.50403
[8]	Shao, Q.-M., Maximal inequalities for partial sums of \(\varrho \)-mixing sequences, Ann. Probab., 23, 948-965 (1995) · Zbl 0831.60028
[9]	Teicher, H., Some new conditions for the strong law, Proc. Nat. Acad. Sci. USA, 59, 705-707 (1968) · Zbl 0211.49103