New inequalities for the values of jumps of discrete distribution functions are obtained. The values of jumps are estimated by linear combinations of a finite number of moments of the distributions involved. The obtained inequalities can be used for statistical estimation of the ranges of values of improbable jumps when the frequencies are zero and are not interesting as estimates. The relation of the proved inequalities to the inequalities for probabilities of unions and Cauchy–Bunyakovski and Hölder’s inequalities is discussed. © Pleiades Publishing, Ltd. 2024.