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.



Ergodic theorem and products
$\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\de}{\delta}$
Let $\Om:=\{-1,1\}^{\N\_0}$ and $\mu:=(\frac12\de\_{-1}+\frac12\de\_1)^{\otimes\N\_0}$, where $\N\_0:=\{0,1,\dots\}$ and $\de\_a$ denotes the Dirac point measure at $a$. For $\om\in\Om$, suppose that $f(\om)=g(\om)=\om\_0$, where $\om\_j:=\om(j)$ for $j\in\N\_0$. For $\om\in\Om$ and $j\in\N\_0$, let $(T\om)\_j:=\om\_{j+1}$, so that $X\_j(\om):=f(T^j\om)=\om\_j$. 


Then $X\_0,X\_1,\dots$ are independent Rademacher random variables, with $P(X\_j=\pm1)=1/2$. Let $S\_n:=\sum\_1^n X\_i X\_{2i}$. 
Then the question becomes the following: is it true that 
\begin{equation}
 A\_n:=\tfrac1n\,S\_n\to EX\_0^2
\end{equation}
almost surely (or at least in probability), as $n\to\infty$?


We have 
\begin{equation}
 E S\_n^2=\sum\_1^n EX\_i^2 X\_{2i}^2+2\sum\_{1\le i<j\le n}EX\_i X\_{2i}X\_j\, EX\_{2j}=n, 
\end{equation}
so that $EA\_n^2=1/n\to0$ and hence $A\_n\to0$ in probability. On the other hand, $EX\_0^2=1\ne0$.  

So, the answer to your question is no in general. 


