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.