The effect of instantaneous nonlinear devices on cross-correlation

If X_1(t), X_2(t) are two noises (stochastic processes), f and g are functions describing the action of two instantaneous nonlinear devices, we say that the (m, n) cross-correlation property holds in case the cross-correlation of f(X_1(t_1)) with g(X_2(t_2)) is proportional to the cross-correlation of X_1(t_2) with X_2(t_2) , whenever f and g are polynomials of degrees not exceeding m and n , respectively. We take m =\infty or n =\infty to mean that f or g is any continuous function. The Barrett-Lampard expansion ^2 of the second-order joint density of X_1(t_1) and X_2(t_2) is used to derive an expression for the cross-correlation of f(X_1(t_1)) and g(X_2(t_2)) . This expression yields necessary and sufficient conditions for the validity of the cross-correlation property in three cases: X_1(t) and X_2(t) stationary, m, n unrestricted; X_1(t) stationary, m, n unrestricted; X_1(t) stationary, n = 1 . Examples are constructed with the help of special orthonormal polynomials illustrating the necessity and sufficiency of the conditions.