Precise lim sup behavior of probabilities of large deviations for sums of i.i.d. random variables (Q1777872)

The classical Cramér-Chernoff-type large deviation principle is proved for the sum \(S_n\) of real valued i.i.d.\ random variables. The random variables are not assumed to satisfy the Cramér condition. It is required that for some \(0<p<2\), \(S_n/n^{1/p}\rightarrow 0\) in probability and that there exists a positive non-increasing function \(\varphi(x)\) on \([0,\infty )\) which is regularly varying with index \(\alpha \leq -1\), such that \(\lim_{x\to \infty} x\varphi (x)=0\) and \(\lim_{x\to\infty}\sup P(| X| >x^{1/p}) / \varphi (x)=1 \). Here \(X\) is a copy of a summand of \(S_n\). Under these conditions it is proved that \[ \lim_{n\to\infty}\sup P(| S_n| > tn^{1/p}) /(n \varphi (n))=t^{p\alpha}. \]