Lower asymptotic relationships for large deviation probabilities (Q1175841)

Let \(X_ i\), \(i\geq 1\), be independent random variables such that \(E X_ i=0\), \(E X_ i^ 2=\sigma^ 2_ i<\infty\), and let \(S_ n=X_ 1+\dots+X_ n\), \(B^ 2_ n=E S^ 2_ n\). Assume that \(E| X_ i|^{2+\delta}<\infty\) for some \(\delta>0\), and put \(c_ n=\max_{i\leq n}(E| X_ i|^{2+\delta'}/\sigma^ 2_ i)^{1/\delta'}\), where \(\delta'=\min(\delta,1)\). The main Theorem 1 of the paper states that if \(0<x_ 0\leq x\leq \alpha B_ n/c_ n\) for a fixed \(\alpha\) and \(x_ 0\), then there are constants \(K>0\) and \(\rho>0\) dependent only on \(\delta'\), \(\alpha\) and \(x_ 0\), such that \[ P[S_ n>xB_ n]/(1-\Phi(x))\geq [1-K(xc_ n/B_ n)^{\delta'}]\exp\{-\rho x^ 2(xc_ n/B_ n)^{\delta'}\}, \] where \(\Phi\) denotes the standard normal distribution function. The case when \(E| X_ i|^{2+\delta}=\infty\) for arbitrarily small \(\delta >0\) is also discussed.