Light tails: Gibbs conditional principle under extreme deviation
Résumé
Let X₁,..,X_{n} denote an i.i.d. sample with light tail distribution and S₁ⁿ denote the sum of its terms; let a_{n} be a real sequence going to infinity with n. In a previous paper ([hal-00813262]) it is proved that as n→∞, given (S₁ⁿ/n>a_{n}) all terms X_{i_{ }} concentrate around a_{n} with probability going to 1. This paper explores the asymptotic distribution of X₁ under the conditioning events (S₁ⁿ/n=a_{n}) and (S₁ⁿ/n≥a_{n}) . It is proved that under some regulatity property, the asymptotic conditional distribution of X₁ given (S₁ⁿ/n=a_{n}) 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 (S₁ⁿ/n≥a_{n}) the dominating point property holds. It also considers the case when the X_{i}'s are R^{d}-valued, f is a real valued function defined on R^{d} and the conditioning event writes (U₁ⁿ/n=a_{n}) or (U₁ⁿ/n≥a_{n}) with U₁ⁿ:=(f(X₁)+..+f(X_{n}))/n and f(X₁) has a light tail distribution. As a by-product some attention is paid to the estimation of high level sets of functions.
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