On some conditions for rationally-infinite divisibility of discrete distributions

New class of rationally-infinitely divisible probability distributions is an essential extension of the funda-mental class of infinitely divisible laws. By definition, a distribution function is called rationally-infinitely divisible if its convolution with some infinitely divisible distribution function is infinitely divisible. The characteristic functions of such distribution functions admit Levy-Khinchine representation, where the spectral function has a bounded total variation on the real line and it can be non-monotonic. Now this new class is investigated and it gets various applications. There are some works devoted to criteria for belonging to this class. The conditions of, in fact, all results are expressed in terms of characteristic functions. In this article, considering only discrete distributions at arbitrary points of the real line, we obtain conditions for the rationally-infinite divisibility of their distribution functions expressed in terms these points and their probabilities. We show that membership in a given class is significantly affected by how many elements among these points are linearly independent over the field of rational numbers.