
QuasiMoment Functions in the Theory of Random Processes
P. I. Kuznetsov^{}, R. L. Stratonovich^{}, V. I. Tikhonov^{} ^{} Moscow
Abstract:
Instead of the system of generalized correlation functions, which statistically describe completely the stochastic process a new system of functions is introduced called quasimoment functions, which also completely describe the process.
The characteristic function of a multidimensional distribution (formula (1.17)) is as easily expressed through quasimoment 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.
Quasimoment functions serve as the coefficients when expanding the probability density of multidimensional distributions in a series (1.20) in Hermite's multidimensional polynomials (a generalized Edgeworth series). Thus, the problem of determining the error committed when breaking off the sequence of quasimoment functions amounts to solving the well known convergence problem of orthogonal polynomial expansions and the accuracy with which the function may be represented by the final series.
By means of quasimoment functions and Hermite’s multidimensional polynomials, it becomes just as simple to write the probability density for distributions of any multiplicity including the continual distribution (a functional probability giving the distribution in a functional space).
It allows solving several important problems, e. g., the problem of nonlinear extrapolation, interpolation and filtration of nonGaussian processes.
The orthogonality of Hermite’s multidimensional polynomials is used to determine the quasimoment functions as ensemble averages from corresponding polynomials.
A derived formula connects quasimoment with correlation functions.
It is shown that with linear and nonlinear transformations of stochastic processes quasimoment functions are transformed linearly.
The transformation of Rayleigh’s stochastic process to a linear system is considered as an example.
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Theory of Probability and its Applications, 1960, 5:1, 80–97
Received: 15.03.1958
Citation:
P. I. Kuznetsov, R. L. Stratonovich, V. I. Tikhonov, “QuasiMoment Functions in the Theory of Random Processes”, Teor. Veroyatnost. i Primenen., 5:1 (1960), 84–102; Theory Probab. Appl., 5:1 (1960), 80–97
Citation in format AMSBIB
\Bibitem{KuzStrTik60}
\by P.~I.~Kuznetsov, R.~L.~Stratonovich, V.~I.~Tikhonov
\paper QuasiMoment Functions in the Theory of Random Processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1960
\vol 5
\issue 1
\pages 84102
\mathnet{http://mi.mathnet.ru/tvp4815}
\transl
\jour Theory Probab. Appl.
\yr 1960
\vol 5
\issue 1
\pages 8097
\crossref{https://doi.org/10.1137/1105007}
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