Abstract
Let X1, X2, . . . be i.i.d. random variables, and set S n =X1+ . . . +X n . Several authors proved convergence of series of the type f(ɛ)=∑ n c n P(|S n |>ɛa n ),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑ n c n P(|S n |>ɛa n ),ɛ>α, is equivalent to the convergence of the double sum ∑ k ∑ n c n P(|S n |>ka n ). Two exceptional series required deriving necessary and sufficient conditions for E[sup n |S n |(logn)η/n]<∞,0≤η≤1.
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Spătaru, A. Strengthening classical results on convergence rates in strong limit theorems. Probab. Theory Relat. Fields 136, 1–18 (2006). https://doi.org/10.1007/s00440-004-0404-5
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DOI: https://doi.org/10.1007/s00440-004-0404-5