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

This paper investigates different classes of stochastic processes (classes T⁽ᵏ⁾ , S⁽ᵏ⁾ ,Φ ⁽ᵏ⁾ , Δ ⁽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 η (t) on the basis of the spectral semi-invariants of the process ξ (t) is given for a large group of non-linear transformations η = Nξ of class Φ ⁽ᵏ⁾ processes (Theorem 2). It is shown that the class Δ(∞ ) is invariant with respect to a large group of non-linear transformations (Theorem 3). Theorem 4 shows that the process η (t) = f(ξ (t - τ )) belongs to the class Δ ⁽²⁾, where ξ (t) ∈ Δ (∞ ) and the functional $f(x(t))$, in the space of trajectories $x(t)$ of the process ξ (t), belongs to a mean square closure of the family of polynomials (3.17).