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

A. N. Shiryaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 3, pp. 293-313

Résumé

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).