Abstract:
A class of multidimensional absolutely continuous distributions is
considered. Each distribution has a moment generating function, which
is finite in a bounded convex set $S$ and generates a family of the
so-called conjugate distributions. We focus our attention on the
limit distributions for this family when the conjugate parameter tends
to the boundary of $S$. As in the one-dimensional case, each limit
distribution is obtained as a corollary of the Abel-type theorem.
The results obtained are utilized for establishing a local limit
theorem for large deviations of arbitrarily high order.
Keywords:Cramér's condition, deviation function, gamma-like distribution, large deviations of arbitrarily high order, local limit theorem, regular variation, support function.