## 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

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