Ergodic theorem and products 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.
 A: These averages converge almost surely, although as the previous comment shows it is not clear how to identify the limit. The almost sure convergence is a theorem of Bourgain from the following reference.
"Double recurrence and almost sure convergence", J. Reine Angew. Math. 
404, pp 140--161, 1990.
It is unknown whether averages of the form $\frac{1}{n} \sum_{k=0}^{n-1} f_1(T^kx)f_2(T^{2k}x)f_3(T^{3k}x)$ converge almost surely.
A: $\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\de}{\delta}$
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. 
