On Large and Superlarge Deviations of Sums of Independent Random Vectors Under Cramér's Condition. II

Abstract

The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255]. In that paper we studied, in the univariate case, the asymptotics of the probabilities that a sum of independent identically distributed random variables will hit a half-interval $[x,x+\Delta)$ in the zone of superlarge deviations when the relative (scaled) deviations ${\alpha=x/n}$ unboundedly increase together with the number of summands n and, at the same time, remain in the analyticity domain of the large deviations rate function for the summands. In the multivariate case, the first part of the paper presented sufficient conditions which ensure that integrolocal and local theorems of the same universal type as in the large and normal deviations zones will also hold in the superlarge deviations zone. The second part of the paper deals with the same problems for three classes on the most wide-spread univariate distributions, for which one can obtain simple sufficient conditions, enabling one to find the asymptotics of the desired probabilities, as $x/n\to \infty$, in the above-mentioned universal form. These are the classes of the so-called exponentially and “superexponentially" fast decaying regular distributions. For them, we also establish limit theorems for the Cramér transforms with parameter values close to the “critical" one. Moreover, we obtain asymptotic expansion for the large deviations rate function.