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On large deviations of sums of independent random variables. (English) Zbl 1125.60024
Summary: Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramér’s condition. The large deviation $$x$$-region under consideration is broader than in the classical Cramér’s theorem, and the estimate of the remainder is uniform with respect to $$x$$. The corresponding asymptotic expansion with arbitrarily many summands is also obtained.

MSC:
 60F10 Large deviations 60G50 Sums of independent random variables; random walks 62E20 Asymptotic distribution theory in statistics
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References:
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