Large deviations of random variables with a finite number of approximately evaluated cumulants

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 $S_n$.