Abstract
Assume that (X n) are independent random variables in a Banach space, (b n) is a sequence of real numbers, Sn= ∑ n1 biXi, and Bn=∑ n1 b 2i . Under certain moment restrictions imposed on the variablesX n, the conditions for the growth of the sequence (bn) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (Sn/B n ln ln Bn)1/2).
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References
N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan,Probability Distributions in Banach Spaces [in Russian], Nauka, Moscow (1985).
A. Araujo and E. Gine,The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York (1980).
J. Kuelbs, “A strong convergence theorem for Banach space valued random variables,”Ann. Probab.,4, No. 5, 744–771 (1976).
V. Goodman, J. Kuelbs, and J. Zinn, “Some results on the LIL in Banach space with applications to weighted empirical processes,”Ann. Probab.,9, No. 5, 713–752 (1981).
A. Acosta, J. Kuelbs, and M. Ledoux, “An inequality for the law of the iterated logarithm,”Lect. Notes Math.,990, 1–29 (1983).
N. H. Bingam, “Variants on the law of the iterated logarithm,”Bull. London Math. Soc,18, No. 5, 433–467 (1986).
V. V. Petrov,Sums of Independent Random Variables [in Russian], Nauka, Moscow (1972).
A. I. Martikainen and V. V. Petrov, “On necessary and sufficient conditions for the law of the iterated logarithm”,Teor. Veroyatn. Ee Primen.,22, Issue 1, 18–26 (1977).
O. I. Klesov, “The law of the iterated logarithm for weighted sums of independent equally distributed random variables”,Teor. Veroyatn. Ee Primen.,31, Issue 2, 389–391 (1986).
R. J. Tomkins, “Lindeberg functions and the law of the iterated logarithm,”Z. Wahrscheinlichkeitstheor. Verw. Geb.,65, No. 1, 135–143 (1983).
I. K. Matsak, “On the summation of independent random variables by the Riesz method”,Ukr. Mat. Zh,44, No. 5, 641–647 (1992).
M. Weiss, “On the law of the iterated logarithm,”J. Math. Mech,8, No. 2, 121 -132 (1959).
J.-P. Kahan,Random Functional Series [Russian translation], Mir, Moscow (1973).
A. I. Martikainen, “On the unilateral law of the iterated logarithm,”Teor. Veroyatn. Ee Primen.,30, Issue 4, 694–705 (1985).
I. K. Matsak and A. N. Plichko, “Khinchin's inequalities and the asymptotic behavior of the sums Sɛnxn on Banach lattices.”Ukr. Mat. Zh,42, No. 5, 639–644 (1990).
I. F. Pinelis and A. I. Sakhanenko, “Remarks on the inequalities for probabilities of large deviations,”Teor. Veroyatn. Ee Primen.,30, Issue 1, 127–131 (1985).
A. I. Sakhanenko, “On the Levi-Kolmogorov inequalities for random variables with values in a Banach space,”Teor. Veroyatn. Ee Primen.,29, Issue 4, 793–799 (1984).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1225–1231, September, 1993.
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Matsak, I.K., Plichko, A.M. On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space. Ukr Math J 45, 1372–1381 (1993). https://doi.org/10.1007/BF01058635
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DOI: https://doi.org/10.1007/BF01058635