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

Juszczak D., Nagaev A. V.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 2

297--320

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.

Multivariate large deviations for sums of i.i.d. random vectors with compactly supported distribution

Zaigraev A.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

315--335

The sums of i.i.d. random vectors with compactly supported and absolutely continuous distribution are considered. Under some conditions the strong form of the local limit theorem for large deviations is proved. In passing the asymptotic behaviour of the moment generating function as well as possible non-degenerate limit laws for the natural exponential family of distributions are established.