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

The present paper continues [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 53 (2009), pp. 301–311] and is devoted to studying the asymptotics of the probability that a sum of independent random vectors falls into a small cube with a vertex at a point $x$ in the large deviation zone. This asymptotics is found in the multivariate case for a class of distributions regularly varying at infinity and for deviations well beyond the Cramér zone.