An Abelian theorem with application to the conditional Gibbs principle

TL;DR

This paper develops an Abelian theorem to approximate moments of tilted distributions and derives a local limit theorem conditioned on extreme deviations for sums of independent, light-tailed variables.

Contribution

It introduces an Abelian theorem for moment approximation and applies Edgeworth expansion to conditioned sums, advancing understanding of extreme deviation behavior.

Findings

Approximate first three moments of tilted distributions.

Derived Edgeworth expansion for normalized sums.

Established a local limit theorem under extreme deviations.

Abstract

Let $X_1,...,X_n$ be $n$ independent unbounded real random variables which have common, roughly speaking, light-tailed type distribution. Denote by $S_1^n$ their sum and by $\pi^{a_n}$ the tilted density of $X_1$, where $a_n \rightarrow\infty$ as $n\rightarrow \infty$. An Abelian type theorem is given, which is used to approximate the first three centered moments of the distribution $\pi^{a_n}$. Further, we provide the Edgeworth expansion of $n$-convolution of the normalized tilted density under the setting of a triangular array of row-wise independent summands, which is then applied to obtain one local limit theorem conditioned on extreme deviation event $(S_1^n/n=a_n)$ with $a_n\rightarrow \infty$.