Upper bounds for cumulants of the sum of multi-indexed random variables

For the $k$th semi-invariant $S_{k}(\xi )$ of the random variable \[ \xi =\sum_{i\in V} \xi _{i}, \] where $V$ is a subset of $\Z^{d}$, we obtain estimates of the form \[ |S_{k}(\xi )|\le (k!)^{1+\gamma } \Delta ^{-(k-2)},\qquad k=3,4,\ldots \] Here $\gamma \ge 0$, $\Delta \ge 1$ are positive variables depending on the rate of growth of the moments of the random variables $\xi _{i}$, $i\in V$, and on their dependence properties. Combined with the results of Lithuanian mathematicians [1], this result makes possible to prove both a normal limit theorem on large deviations and an estimate for a tail of the distribution of a generalized $U$-statistic.