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).

Full version