Estimating the Kernels of Nonlinear Orthogonal Polynomial Functionals

Abstract

Let $(X(t), Y(t))$ be a complex vector process stationary of order $k$ for any $k, k = 1,2,\cdots$, such that $Y(t)$ is expressed as a polynomial functional of degree 2 operating on $X(t)$. Then $Y(t)$ can be rewritten as a sum of orthogonal projections $G_j(K_j, Y(t)), j = 0, 1, 2$. It is shown that there is a set of functionals which approximate in mean square the projection $G_2(K_2, Y(t))$. Moreover, it is possible to determine the kernels associated with these functionals.