Products of independent wss stochastic processes

Let {X(t)}_t∈R
and {Y (t)}_t∈R
be two centered wss stochastic processes of respective covariance functions C_X
(τ ) and C_Y (τ ).

1. Assume the two signals to be independent. Show that Z (t)
:= X(t)Y (t) (t ∈
R) is a wss stochastic process. Give its mean and covariance function.

2. Assume the same hypothesis as in the previous question,
but now {X(t)}t∈R
is the harmonic process of Example 5.1.20. Suppose that {Y (t)}_t∈R
admits a power spectral density f_Y (ν). Give the power spectral
density f_Z (ν) of {Z (t)}_t∈R .