If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that

$$ \lim_{n \rightarrow \infty} \frac{f_n}{n} = \int_{\Omega} f_1(\omega) d\mu. $$

I was wondering what happens if one studies $h_n (\omega) = \sum_{i=1}^n f (T^i \omega)g(T^{2i} \omega)$ is it still true that

$$ \lim_{n \rightarrow \infty} \frac{h_n}{n} = \int_{\Omega} f(\omega)g(\omega) d\mu? $$

If we assume that $g(T^{2i}\omega)$ form a family of i i d random variables.