Abstract
The series\(\sum\nolimits_{n \geqslant 1} {\tau _n P(|S_n | \geqslant \varepsilon n^a )}\) is studied, where Sn are the sums of independent equally distributed random variables, τn is a sequence of nonnegative numbers, α>0, and ɛ>0 is an arbitrary positive number. For a broad class of sequences τn, the necessary and sufficient conditions are established for the convergence of this series for any ɛ>0.
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References
P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,”Proc. Nat. Acad. Sci. U.S.A,33, No. 2, 25–31 (1947).
M. Katz, “The probability in the tail of a distribution,”Ann. Math. Statist.,34, No. 1, 312–318 (1963).
C. C. Heyde and V. K. Rohatgi, “A pair of complementary theorems on convergence in the law of large numbers,”Proc. Cambr. Philos. Soc.,63, No. 1, 73–82 (1967).
S. Kh. Sirazhdinov and M. U. Gafurov,Method of Series in the Limit Problems for Random Walks [in Russian], Fan, Tashkent (1987).
S. Asmussen and T. G. Kurtz, “Necessary and sufficient conditions for complete convergence in the law of large numbers,”Ann. Probab.,8, No. 1, 176–182 (1980).
R. T. Smythe, “The sums of independent random variables on the partially ordered sets,”Ann. Probab.,2, No. 5, 902–917 (1974).
A. Gut, “Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices,”Ann. Probab.,6, No. 3, 469–482 (1978).
O. I. Klesov, “The strong law of large numbers for multiple sums of independent equally distributed random variables,”Mat. Zametki,38, No. 6, 915–230 (1985).
C. C. Heyde, “A supplement to the strong law of large numbers,”J. Appl. Probab.,12, 173–175 (1975).
V. V. Petrov,The Sums of Independent Random Variables [in Russian], Nauka, Moscow (1972).
Y. S. Chow and H. Teicher,Probability Theory, Springer, New York (1978).
J. Hoffman-Jorgensen, “Sums of independent Banach space valued random variables,”Studia. Math.,54, 159–186 (1974).
B. von Bahr and C.-G. Esséen, “Inequalities for ther-th absolute moment of a sum of random variables, 1≤r≤2”,Ann. Math. Statist.,36, No. 1, 299–303 (1965).
E. Seneta,Regularly Varying Functions [in Russian], Nauka, Moscow (1985).
M. K. Kholmuradov, “The law of the iterated logarithm for sums of random variables with many-dimensional indices,”Dokl. Akad. Nauk Uzbek SSR, No. 7, 3–4 (1985).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 770–784, June, 1993.
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Klesov, O.I. Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables. Ukr Math J 45, 845–862 (1993). https://doi.org/10.1007/BF01061437
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DOI: https://doi.org/10.1007/BF01061437