I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way.
Is it possible for the time averages of certain functions of a non wide-sense stationary process (NWSS) to converge to their respective ensemble average under mean square measure?
For instance, suppose $X(t)$ is the NWSS process in question, which has ensemble average $E\left[X(t)\right]=\mu(t)$ and $\mu(t)$ is known and satisfies $\mu^{-1}(t)\neq0$ $\forall$ $t\geq0$. Clearly, $X(t)$ is not mean-ergodic (since the ensemble average depends on $t$, whereas the time average cannot).
But consider the random quantity $\mu^{-1}(t)X(t)$, can this converge to its ensemble average under the measure of mean-square? If so, how can I determine if the quantity is ergodic, i.e. if: \begin{equation} \lim_{T\to\infty}\frac{1}{T}\int_0^T\mu^{-1}(t)X(t)dt\to 1\;\text{?} \end{equation}
Any help is greatly appreciated (and it would be great to have a reference that I could check out given my lack of knowledge in stochastic processes!)