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 Teor. Veroyatnost. i Primenen., 1960, Volume 5, Issue 3, Pages 293–313 (Mi tvp4836)

Some Problems in the Spectral Theory of Higher-Order Moments. I

A. N. Shiryaev

V. A. Steklov Mathematical Institute, USSR Academy of Sciences

Abstract: This paper investigates different classes of stochastic processes (classes $\mathbf T^{(k)}$, $\mathbf S^{(k)}$,$\mathbf\Phi^{(k)}$, $\mathbf\Delta^{(k)}$, which are defined in the introduction) by examining their high-order spectral moments and semi-invariants.
The paper considers linear (see Theorem 1 for example) and non-linear transformations of stochastic processes. A formula for determining spectral semi-invariants of the process $\eta(t)$ on the basis of the spectral semi-invariants of the process $\xi(t)$ is given for a large group of non-linear transformations $\eta=N\xi$ of class $\mathbf\Phi^{(k)}$ processes (Theorem 2).
It is shown that the class $\mathbf\Delta^{(\infty)}$ is invariant with respect to a large group of non-linear transformations (Theorem 3). Theorem 4 shows that the process $\eta(t)=f(\xi(t-\tau))$ belongs to the class $\mathbf\Delta^{(2)}$, where $\xi(t)\in\mathbf\Delta^{(\infty)}$ and the functional $f(x(t))$, in the space of trajectories $x(t)$ of the process $\xi (t)$, belongs to a mean square closure of the family of polynomials (3.17).

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English version:
Theory of Probability and its Applications, 1960, 5:3, 265–284

Citation: A. N. Shiryaev, “Some Problems in the Spectral Theory of Higher-Order Moments. I”, Teor. Veroyatnost. i Primenen., 5:3 (1960), 293–313; Theory Probab. Appl., 5:3 (1960), 265–284

Citation in format AMSBIB
\Bibitem{Shi60} \by A.~N.~Shiryaev \paper Some Problems in the Spectral Theory of Higher-Order Moments.~I \jour Teor. Veroyatnost. i Primenen. \yr 1960 \vol 5 \issue 3 \pages 293--313 \mathnet{http://mi.mathnet.ru/tvp4836} \transl \jour Theory Probab. Appl. \yr 1960 \vol 5 \issue 3 \pages 265--284 \crossref{https://doi.org/10.1137/1105026} 

• http://mi.mathnet.ru/eng/tvp4836
• http://mi.mathnet.ru/eng/tvp/v5/i3/p293

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This publication is cited in the following articles:
1. A. M. Stepin, “Spectral properties of generic dynamical systems”, Math. USSR-Izv., 29:1 (1987), 159–192