Asymptotics of the rate function in the large deviation principle for sums of independent identically distributed random variables

Abstract

Let ${\mathrm{\Lambda }}^{\ast }$ be the rate function in the large deviation principle for the sums $X_1+\cdots +X_n$ of independent identically distributed random variables $X_1,X_2,\dots$. It is shown that ${\mathrm{\Lambda }}^{\ast }(x)\sim -ln\mathsf{P}(X_1\ge x)$ (as $x\to \mathrm{\infty }$) if and only if $ln\mathsf{P}(X_1\ge x)\sim L_0(x)$ for some concave function $L_0$. The main ingredient of the proof is the general, explicit expression of a suitable quasi-minimizer in $t\ge 0$ of the Bernstein–Chernoff upper bound $e^{-tx}\mathsf{E}{e}^{t{X}_1}$ on $\mathsf{P}(X_1\ge x)$, which is amenable to analysis and, at the same time, is close enough to a true minimizer.