\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\P}{\mathcal{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\d}{\mathsf{d}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \newcommand{\B}{\mathscr{B}} \newcommand{\fou}{\mathcal{F}} \]

Put $X = \{0,1\}^\N$. Write $\mu_p$ for the $(p,1-p)$ coin measure.

1. What is $\mu_p([01])$?
2. What is $\displaystyle\int 1_{[01]} \intd \mu_p$?
3. Define $Z_1(x) = x(1) x(2)$. What is $\displaystyle\int Z_1 \intd \mu_p$?
4. What is $\mu_p(T^{-1}([01]))$?
5. Define $Z_2 = Z_1 \circ T$. What is $\displaystyle \int Z_2 \intd \mu_p$?

For $n \ge 2$ define $Z_n = Z_1 \circ T^{n-1}$. Define \[ Y_n = Z_n - \int Z_n \intd \mu_p \] for all $n \in \N$.

6. Calculate $\displaystyle\int Y_n \intd \mu_p$ for all $n \in \N$.
7. Calculate $\displaystyle\int Y_n Y_{n+2}\intd \mu_p$ for all $n \in \N$.
8. Calculate $\displaystyle\int Y_n Y_{n+1}\intd \mu_p$ for all $n \in \N$.
9. For which $a,b,c,d \in \N$ is $\displaystyle\int Y_a Y_b Y_c Y_d \intd \mu_p$ equal to zero?
10. Prove the complement of \[ \left\{ x \in X : \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x(n) x(n+1) \textsf{ exists} \right\} \] has zero measure with respect to $\mu_p$ and identify the limit.