On Large and Superlarge Deviations of Sums of Independent Random Vectors under Cramér's Condition. I

Abstract

We study the asymptotics of the probability that the sum of independent identically distributed random vectors is in a small cube with a vertex at point x in the following two problems. (A) When the relative (normalized) deviations $x/n$ (n is the number of terms in the sum) are in the analyticity domain of the large deviation rate function $\Lambda(\alpha)$ for the summands (if, in addition, $|x|/n\to\infty$, then one speaks of superlarge deviations). (B) When the alternative possibility takes place, i.e., when $x/n$ is outside the analyticity domain of the function $\Lambda(\alpha)$. In problems (A) and (B) the asymptotics of the superlarge deviation probabilities (when $|x/n|\to\infty$), just as the asymptotics of the probabilities of the “usual large deviation in problem (B) (when $x/n$ is bounded away from the expectation of the summands and remains bounded), in many aspects remained unknown. The present paper, consisting of two parts, is mostly devoted to solving problem (A) for superlarge deviatio...