On an arithmetical property of moments and cumulants

Abstract

The main result of the paper is the following. Let a non-degenerate distribution have finite moments ${\mu }_k$ of all orders $k=0,1,2,\dots$. Then the sequence $\{{\mu }_k/k!,k=0,1,2,\dots \}$ either contains infinitely many different terms or at most three. In the latter case, this sequence has the form $\{1,a,1-b,a,1-b,a,1-b,\dots \}$ and corresponds to a distribution with the characteristic function \begin{equation*}\label{ eq0} f(t)=\frac{1+iat+bt^2}{1+t^2}, \quad \text{where} \;\; b\geq 0,\; \frac{1-a-b}{2}\geq 0,\; \frac{1+a-b}{2}\geq 0. \end{equation*}.