Light tails: Gibbs conditional principle under extreme deviation

الأصل · EN

Let X₁,..,Xₙ denote an i.i.d. sample with light tail distribution and S₁ⁿ denote the sum of its terms; let aₙ be a real sequencegoing to infinity with n.In a previous paper (BoniaCao) it is proved that as n→∞, given (S₁ⁿ/n>aₙ) all terms Xᵢ concentrate around aₙ with probability going to 1. This paper explores the asymptotic distribution of X₁ under the conditioning events (S₁ⁿ/n=aₙ) and (S₁ⁿ/n≥ aₙ). It is proved that under some regulatity property, the asymptotic conditional distribution of X₁ given (S₁ⁿ/n=aₙ) can be approximated in variation norm by the tilted distribution at point aₙ, extending therefore the classical LDP case developed in Diaconis and Freedman (1988). Also under (S₁ⁿ/n≥ aₙ) the dominating point property holds. It also considers the case when the Xᵢ's are Rᵈ-valued, f is a real valued function defined on Rᵈ and the conditioning event writes (U₁ⁿ/n=aₙ) or (U₁ⁿ/n≥ aₙ) with U₁ⁿ:=(f(X₁)+..+f(Xₙ)) /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.