Geodesic
Abelian theorems, limit properties of conjugate distributions,
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 701-719

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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. 
@article{TVP_2003_48_4_a3,
     author = {A. Yu. Zaigraev and A. V. Nagaev},
     title = {Abelian theorems, limit properties of conjugate distributions,},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {701--719},
     publisher = {mathdoc},
     volume = {48},
     number = {4},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a3/}
}
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A. Yu. Zaigraev; A. V. Nagaev. Abelian theorems, limit properties of conjugate distributions,. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 701-719. http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a3/