Relations between cumulants and central moments and their applications

Abstract

At the beginning of this paper, we consider the solution to the problem of constructing relations between mixed central moments and cumulants (semi-invariants) of an arbitrary vector random variable. The sought relations are derived on the basis of formal operations over the Maclaurin series, which are various expansions of the characteristic function of the random vector. In the case under consideration, the coefficients of the expansions are mixed moments, cumulants, and central moments. One of the applications for the recurrence relations obtained is their usage to closure systems of ordinary differential equations (ODE) for the functions of the mathematical expectations and the functions of central moments until a given order. These functions are the main probabilistic speci cations for the state vectors of systems of stochastic ordinary differential equations (SODE) describing a behavior of stochastic dynamic systems. Therefore we have devoted the last part of the paper to derivation of the ODE system satisfied by the indicated moment functions.