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 $n\to \infty$, given $\left(S_1^n/n>a_n\right)$ all terms $X_{i_{}}$ 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 $\left(S_1^n/n\ge a_n\right)$ . 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 $\left(S_1^n/n\ge a_n\right)$ 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 $\left(U_1^n/n\ge a_n\right)$ 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.

arXiv admin note: substantial text overlap with arXiv:1206.6951, arXiv:1302.1337

References in corpus (1)

• A conditional limit theorem for random walks under extreme deviation