Probability Theory and Mathematical Statistics : Proceedings of the Sixth Vilnius Conference, Vilnius, Lithuania, 28 June{u2013}3 July, 1993

Let \(X\) be a random variable (r.v.), assuming non-negative integer values, \(E X=\lambda>0\) and the factorial cumulants \(\widetilde{\Gamma}_{k}(X)\) of which satisfy some conditions of growth. Then under the condition ( \(\widetilde{\mathbf{S}}\) ) (see below) a general lemma (Lemma 1) is proved comparing the behaviour of probabilities of large deviations \(P\{X \geqslant x\}\) of a r.v. \(X\) against Poisson distribution \(P\{\eta \geqslant x\}\), where \(\eta\) is a Poisson r.v. with parameter \(\lambda>0\) and \(\lambda