Light tails: Gibbs conditional principle under extreme deviation

Abstract

Let $X_{1},..,X_{n}$ denote an i.i.d. sample with light tail distribution and $S_{1}^{n}$ denote the sum of its terms; let $a_{n}$ be a real sequence\ going to infinity with $n.$\ In a previous paper (\cite{BoniaCao}) it is proved that as $n\rightarrow\infty$, given $\left(S_{1}^{n}/n>a_{n}\right) $ all terms $X_{i_{\text{}}}$ concentrate around $a_{n}$ with probability going to 1. This paper explores the asymptotic distribution of $X_{1}$ under the conditioning events $\left(S_{1}^{n}/n=a_{n}\right) $ and $\left(S_{1}^{n}/n\geq a_{n}\right)$ . It is proved that under some regulatity property, the asymptotic conditional distribution of $X_{1}$ given $\left(S_{1}^{n}/n=a_{n}\right) $ can be approximated in variation norm by the tilted distribution at point $a_{n}$, extending therefore the classical LDP case developed in Diaconis and Freedman (1988) . Also under $\left(S_{1}^{n}/n\geq a_{n}\right) $ the dominating point property holds. It also considers the case when the $X_{i}$'s are $\mathbb{R}^{d}-$valued, $f$ is a real valued function defined on $\mathbb{R}^{d}$ and the conditioning event writes $\left(U_{1}^{n}/n=a_{n}\right) $ or $\left(U_{1}^{n}/n\geq a_{n}\right)$ with $U_{1}^{n}:=\left(f(X_{1})+..+f(X_{n})\right) /n$ and $f(X_{1})$ has a light tail distribution$.$ As a by-product some attention is paid to the estimation of high level sets of functions.