Asymptotic expansions for probabilities of large deviations for sums of independent random variables in the case of violation of Cramer’s condition. (Russian) Zbl 0576.60021

Vinogradov, V. V.

Teor. Veroyatn. Mat. Stat. 31, 19-25 (1984).

Let \(X_ 1,X_ 2,..\). be independent and identically distributed random variables with a polynomial upper tail \(P\{X_ 1>x\}=\sum^{k}_{i=1}c_ ix^{-\alpha_ i}+o(x^{-\alpha_ 1- q})\), as \(x\to \infty\), where \(c_ 1,...,c_ k>0\), \(2<\alpha_ 1<...<\alpha_ k\leq \alpha_ 1+q\), \(q>0\), and assume \(EX_ 1=0\) and \(E| X_ 1|^ r<\infty\) for some \(2<r<\min (\alpha_ 1,3)\). The author derives the asymptotic expansion of the form \[ P\{X_ 1+...+X_ n>x\}=n\{P\{X_ 1>x\}+\sum^{k}_{i=1}c_ ig(n,r,c_ i,x\quad,\epsilon)x^{-\alpha_ i}\}+o(n^{(5-r)/2}x^{-\alpha_ 1- 1})+o(nx^{-\alpha_ 1-q}), \] as \(n\to \infty\), valid for \(x\geq n^{1/2+\epsilon}\) with an arbitrary \(\epsilon >0\), where the coefficient function g is explicitly given.

Reviewer: S.Csörgö

MSC:

60F10

Large deviations

Keywords:

polynomial upper tail; asymptotic expansion