Large-deviation probability and the local dimension of sets

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Let X1, X2, ..., Xn be n independent identically distributed real random variables and Sn = Σn=1n Xi. We obtain precise asymptotics forP (Sn ∈ nA) for rather arbitrary Borel sets A1 in terms of the density of the dominating points in A. Our result extends classical theorems in the field of large deviations for independent samples. We also obtain asymptotics forP (Sn ∈ γnA), with γn/n → ∞.