The asymptotic behavior of one-sided large deviation probabilities. I (Q1366383)

The author discusses conditions upon a distribution function \(F\) under which there exists a sequence \((t_n)_{n=1,2,\dots}\) such that the distribution function \(F_n\) of the \(n\)-fold convolution satisfies the asymptotic relation \((1- F_n(x))\sim n(1- F(x))\), \(n\to\infty\), uniformly for all \(x\), \(x\geq t_n\). Furthermore, the condition \(t_n= O(n)\) is refined taking into account the rate of decrease of the tails \(1-F(x)\) as \(x\to\infty\).