Local large deviation theorem for sums of i.i.d. random vectors when the Cramér condition holds in the whole space

A class of multidimensional distributions is considered. This class contains all the elliptically contoured distributions having sup-exponential weight function. Each representative of the class determines a family of the so-called exponential or conjugate distributions. It is established that the conjugate distribution is asymptotically normal. On the basis of this normality a large deviation local limit theorem is proved. The theorem assumes no restrictions on the order of deviations.