  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.
