Abstract
We make some remarks leading to a refinement of the recent work of Klesov (1993) on the connection between the convergence of the series \(\Sigma _{n = 1}^\infty \tau _n P(|S_n | \ge \varepsilon n^\alpha )\) for every ε > 0 and the convergence of \(\Sigma _{n = 1}^\infty n\tau _n P(|X_1 | \ge \varepsilon n^\alpha )\) again for every ε > 0.
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References
P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,” Proc. National Acad. Sci. USA, 33, No. 2, 25–31 (1947).
P. Erdós, “On a theorem of Hsu and Robbins,” J. Math. Statist., 20, 286–291 (1949).
O. I. Klesov, “Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables,” Ukr. Mat. Zh., 45, No. 6, 770–784 (1993).
M. Loeve, Probability Theory, Van Nostrand, New York 1955.
V. V. Petrov, Limit Theorems for Sums of Independent Random Variables [in Russian], Nauka, Moscow 1987.
E. Giné and J. Zinn, “Central limit theorems and weak laws of large numbers in certain Banach spaces,” Z. Wahrscheinlichtkeits-theor. und verw. Geb., 62, 323–354 (1983).
M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, Springer, Berlin 1991.
S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhauser, Boston 1992.
A. R. Pruss, “Randomly sampled Riemann sums and complete convergence in the law of large numbers for a case without identical distribution,” Proc. Amer. Math. Soc. (to appear).
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Pruss, A.R. Remarks on summability of series formed of deviation probabilities of sums of independent identically distributed random variables. Ukr Math J 48, 631–635 (1996). https://doi.org/10.1007/BF02390624
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DOI: https://doi.org/10.1007/BF02390624