On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. I

The present paper, consisting of two parts, is sequential to [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255 and pp. 567–594], [A. A. Borovkov and K. A. Borovkov, Theory Probab. Appl., 46 (2002), pp. 193–213 and 49 (2005), pp. 189–206], and [A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks. I. Slowly Decreasing Distributions of Jumps, Nauka, Moscow (in Russian), to be published] and is devoted to studying the asymptotics of the probability that the sum of the independent random vectors is in a small cube with the vertex at point $x$ in the large deviations zone. The papers [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255 and pp. 567–594] are mostly devoted to the “regular deviations” problem (the problem [A] using the terminology of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255 and pp. 567–594]), when the relative (“normalized”) deviations $x/n$ ($n$ is the number of terms in the sum) are in the analyticity domain of the large deviations rate function for the summands (the so-called Cramér deviations zone) and at the same time $|x|/n\to\infty$ (superlarge deviations). In the present paper we study the “alternative” problem of “irregular deviations” when $x/n$ either approaches the boundary of the Cramér deviations zone or moves away from this zone (the problem [B] using the terminology of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255 and pp. 567–594]). In this case the large deviations problems in many aspects remained unknown. The desired asymptotics for deviations close to the boundary of the Cramér zone is obtained in section I of this paper under quite weak conditions in the general multivariate case. Furthermore, in the univariate case we also study the deviations which are bounded away from the Cramér zone. In this case we require some additional regularity properties for the distributions of the summands.