A Gibbs Conditional theorem under extreme deviation

TL;DR

This paper investigates the behavior of i.i.d. sample distributions under extreme sum exceedances, revealing thresholds for asymptotic independence and extending classical large deviation results using advanced Edgeworth expansions.

Contribution

It introduces a new Edgeworth expansion for tilted distributions with diverging parameters, extending the understanding of conditional distributions under extreme deviations.

Findings

Identifies thresholds for asymptotic independence of summands

Extends classical large deviation results to extreme exceedance regimes

Develops a new Edgeworth expansion for specific triangular arrays

Abstract

We explore some properties of the conditional distribution of an i.i.d. sample under large exceedances of its sum. Thresholds for the asymptotic independance of the summands are observed, in contrast with the classical case when the conditioning event is in the range of a large deviation. This paper is an extension to [7]. Tools include a new Edgeworth expansion adapted to specific triangular arrays where the rows are generated by tilted distribution with diverging parameters, together with some Abelian type results.