Abstract: This paper examines the moments of probability distributions, presenting new theorems and their implications. The main result of the paper is the following. Let a nondegenerate distribution have finite moments \mu_k of all orders k=0,1,2,\ldots Then the sequence \{\mu_k/k!\colon k=0,1,2,\ldots\} 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, \ldots\} and corresponds to a distribution with the characteristic function \begin{equation*}\label{ eq0} f(t)=\frac{1+ {\rm i}at+bt^2}{1+t^2}, \qquad \text{where } b\geq 0, 1-a-b \geq 0, 1+a-b \geq 0. \end{equation*} Corresponding distribution is mixture of an atom at zero, exponential distribution on positive semiaxis and exponential distribution on negative semiaxis with weights b, (1+a-b)/2, (1-a-b)/2.
Keywords: classical problem of moments; exponential distribution; Laplace distribution; cumulant; Marcinkiewicz theorem; analytical continuation
DOI: DOI 10.14712/1213-7243.2026.002
AMS Subject Classification: 44A60 30E05 30B40 60E05 60E10