Probabilities of large deviations of the sums of lattice random vectors when the original distribution has heavy tails

The large deviation problem for sums of independent identically distributed random vectors with values from $\mathbb Z^d$ is considered. It is assumed that the underlying distribution has the heavy tails. The special attention is paid to the role which is played by the density of the distribution support at infinity. The proven local and integral theorems generalize until now known results concerned with heavy-tailed distributions.
This work was supported by Komitet Badań Naukowych, grant PB 591/P03/95/08.