On the probabilities of moderate deviations for combinatorial sums

We investigate the asymptotic behavior of the probabilities of moderate deviations for combinatorial sums. We find a zone in which tails of distributions of combinatorial sums have the same asymptotic behavior as that of the standard normal law. The combinatorial sums have dependent increments. It follows from this that the classical method of characteristic functions cannot be applied. We use Esseen-type bounds for combinatorial sums that were recently obtained in another paper by the author. We show that the zone of the normal convergence is close to the best one, which is for the case of centered independent random variables. We consider the case of finite variations of summands. The case of infinite variations is also discussed. © 2015, Allerton Press, Inc.