Strengthening classical results on convergence rates in strong limit theorems |
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Authors: | Aurel Spătaru |
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Affiliation: | (1) Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie No 13, 761 00 Bucharest 5, Romania |
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Abstract: | Let X1, X2, . . . be i.i.d. random variables, and set Sn=X1+ . . . +Xn. Several authors proved convergence of series of the type f(ɛ)=∑ncnP(|Sn|>ɛan),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑ncnP(|Sn|>ɛan),ɛ>α, is equivalent to the convergence of the double sum ∑k∑ncnP(|Sn|>kan). Two exceptional series required deriving necessary and sufficient conditions for E[supn|Sn|(logn)η/n]<∞,0≤η≤1. |
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Keywords: | Primary 60G50 60F10 60G40 secondary 60E15 |
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