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Precise large deviations for sums of random variables with consistently varying tails. (English) Zbl 1051.60032

The authors prove the following main result: Let \(\{X_k,\,k\geq 1\}\) be a sequence of independent, identically distributed nonnegative random varibles with common distribution function \(F\) and finite expectation. Then, for fixed \(\gamma> 0\), \(P(S_n- n\mu> x)\sim n\overline F(x)\) uniformly for \(x\geq \gamma n\).

MSC:

60F10 Large deviations
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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