### PRECISE ASYMPTOTICS OVER A SMALL PARAMETER FOR A SERIES OF LARGE DEVIATION PROBABILITIES

**V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH**

*Theory of Stochastic Processes*

*Vol.13 (29), no.1-2, 2007, pp.44-56*

We obtain the asymptotics of the series

$$

\sum\limits_{k=1}^\infty w_k\mathbf{P}(|S_k|\geq \varepsilon \varphi_k)

$$

as $\varepsilon\downarrow 0$, where $S_k$ are partial sums of independent and identically distributed random variables in the domain of attraction of a nondegenerate stable law, $w$ and $\varphi$ are regularly varying functions (in Karamata’s sense).

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