Let $ \{X(t); t\geq 0\} $ be an ergodic Markov process, and let G be a positive integrable function with $\int_0^\infty G(x)dx=1$. Does $$F(t)\doteq G*X(t) = \int_0^t G(t-s)X(s)ds $$ converge in distribution?
My intuition is that F(t) is a sliding average of the ergodic process $X$, so therefore, it should converge. A discrete-time version is equally desirable.
Thank you!