Geodesic
Large deviations of random variables with a finite number of approximately evaluated cumulants
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 3, pp. 603-608

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The paper establishes a theorem on exact asymptotics of probabilities of large deviations for random variables with known estimates for only a finite number of cumulants, the latter being subject to conditions of simultaneous growth. For instance, let $S_n$ be a sequence of real random variables and assume the existence of a sequence of small in a sense random variables $G_n(\xi)$ depending on $\xi$ analytically and such that $$ \mathsf{E}\exp(\xi S_n+G_n(\xi))=\exp\sum_{j=2}^m\frac{\Gamma_{nj}}{j!}\xi^j. $$ If all the cumulants $\Gamma_{nj}$ have order $n$ and the order of $G_n(\xi)$ does not exceed $n\xi^{m+1}$, then the Cramér type probabilities of large deviations can be indicated for $S_n$.
Mots-clés : random variables, cumulant
Keywords: distribution function, large deviations, Cramer asymptotics. 
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     author = {V. I. Bakhtin},
     title = {Large deviations of random variables with a finite number of approximately evaluated cumulants},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {603--608},
     publisher = {mathdoc},
     volume = {42},
     number = {3},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_3_a10/}
}
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V. I. Bakhtin. Large deviations of random variables with a finite number of approximately evaluated cumulants. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 3, pp. 603-608. http://geodesic.mathdoc.fr/item/TVP_1997_42_3_a10/