Light tails: Gibbs conditional principle under extreme deviation

TL;DR

This paper investigates the asymptotic behavior of i.i.d. light-tailed random variables under extreme sum deviations, showing that their conditional distributions can be approximated by tilted distributions, extending classical large deviation principles.

Contribution

It extends the Gibbs conditional principle to extreme deviations for light-tailed distributions, providing approximation results for the conditional distribution of individual variables.

Findings

Conditional distribution approximates tilted distribution at the dominating point.

Under sum deviations, the dominating point property holds.

Extension to multivariate functions and high-level set estimation.

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.