About stationary and wide-sense stationary processes

Question

I have just started with stochastical calculus, and I need some help with a pair of problems:

$\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the processes $X(t)$ and $X' (t)$ are orthogonal.

$\bullet$If $X(t)$ is a twice mean square differentiable, stationary and Gaussian stochastical process, such that $E[X(t)] = 0$, then $X(t)$ is independent of $X' (t)$ but not independent of $X''(t)$

I was given the hint that I should use the formula $\Gamma_{X^{(n)},X^{(m)}}(t,s)=(-1)^m\frac{d^{(n+m)}\Gamma_X(\tau)}{d\tau^{(n+m)}}$, where $\tau=t-s$, but I can't get anything from it. I'm stuck. Any ideas?

Thanks a lot for any help.

Answer 1

Hint for the first point: For every $t$, consider $m(t)=E[X(t)]$, $n(t)=E[X'(t)]$, $f(t)=E[(X(t)-m(t))^2]$ and $g(t)=E[(X(t)-m(t))(X'(t)-n(t))]$. One knows that the function $m$ and $f$ are constant, and one tries to show that $g(t)=0$. One is also advised to use some derivative of some sort.

Thus, one could start by showing that $n(t)=0$ for every $t$, then express $g(t)$ as a derivative of something involving all the rest...