Mutual information in stationary channels with additive noise

A continuous time stationary channel with additive noise presented by
Y(t) = X(t) + Z(t)
is considered, where
\{X(t)\}
and
\{Z(t)\}
are mutually independent stationary processes representing the channel input and the noise, respectively, and
\{Y(t)\}
represents the channel output. It is shown that, under some general assumptions, the mutual information between the input
\{ X(t); 0 \leq t \leq T \}
and the output
\{ Y(t); 0 \leq t \leq T \}
can be expressed in the form
AT + B(T)
with a constant
A
and a function
B(T)
given in terms of the conditional mutual informations between the "past" and the "future" given the "present" of the output and the noise. It is also shown that the remainder term
B(T)
converges to a constant as
T
tends to infinity.