Quasi-Moment Functions in the Theory of Random Processes

Instead of the system of generalized correlation functions, which statistically describe completely the stochastic process a new system of functions is introduced called quasi-moment functions, which also completely describe the process.
The characteristic function of a multi-dimensional distribution (formula (1.17)) is as easily expressed through quasi-moment as through correlation functions. This characteristic function is represented as the product of two factors; the first factor gives the characteristic function for a given Gaussian process with about the same mean value and correlation function as for the initial process, while the second factor (represented as a series) accounts for the deviation of the characteristic function from the Gaussian. Moreover, probability densities of different multiplicities may be written through the introduced functions. This is important in solving problems connected with condition probability and others.
Quasi-moment functions serve as the coefficients when expanding the probability density of multi-dimensional distributions in a series (1.20) in Hermite's multi-dimensional polynomials (a generalized Edgeworth series). Thus, the problem of determining the error committed when breaking off the sequence of quasi-moment functions amounts to solving the well known conver-gence problem of orthogonal polynomial expansions and the accuracy with which the function may be represented by the final series.
By means of quasi-moment functions and Hermite’s multi-dimensional polynomials, it becomes just as simple to write the probability density for distributions of any multiplicity in-cluding the continual distribution (a functional probability giving the distribution in a functional space).
It allows solving several important problems, e. g., the problem of non-linear extrapolation, interpolation and filtration of non-Gaussian processes.
The orthogonality of Hermite’s multi-dimensional polynomials is used to determine the quasi-moment functions as ensemble averages from corresponding polynomials.
A derived formula connects quasi-moment with correlation functions.
It is shown that with linear and non-linear transformations of stochastic processes quasi-moment functions are transformed linearly.
The transformation of Rayleigh’s stochastic process to a linear system is considered as an example.