A Gibbs conditional theorem under extreme deviation

We explore some properties of the conditional distribution of an independently and identically distributed (i.i.d.) sample under large exceedances of its sum. Thresholds for the asymptotic independence 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 of Broniatowski and Cao [Extremes, 17 (2014), pp. 305–336]. Tools include a new Edgeworth expansion adapted to specific triangular arrays, where the rows are generated by tilted distribution with diverging parameters, and some Abelian type results.