New large-deviation local theorems for sums of independent and identically distributed random vectors when the limit distribution is α-stable

Abstract

A class of absolutely continuous distributions in R^d is considered. Each distribution belongs to the domain of normal attraction of an α-stable law. The limit law is characterized by a spectral measure which is absolutely continuous with respect to the spherical Lebesgue measure. The large-deviation problem for sums of independent and identically distributed random vectors when the underlying distribution belongs to that class is studied. At the focus of attention are the deviations in the directions, where the spectral density equals zero. The main conclusion is that the deviation in such a direction is explained by two abnormally large summands.