On large and superlarge deviations for sums of independent random vectors under the Cramer condition. I

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 super-large 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 super-large 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 super-large deviations. In part I we present a solution to problem (A) in the general multivariate case. As the first step, we use the Cramér transform, which enables one to reduce the problem on super-large deviations of the original sum to that on normal deviations of the sum of the transformed random vectors. Then we use integrolocal or local theorems for sums of random vectors in the triangular array scheme in the normal deviations zone. The required versions of such theorems are contained in [A. A. Borovkov and A. A. Mogulskii, Math. Notes, 79 (2006), pp. 468–482] and in section 5. We also present in part I a scheme for solving problem (B), to which a separate paper will be devoted. In the case when the distribution of the sum is absolutely continuous in a neighborhood of the point $x$, we study the asymptotics of the respective density at that point.