State Birkhoff's almost-everywhere ergodic theorem.

Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables such that

$$\mathbb{P}\left(X_{n}=0\right)=\mathbb{P}\left(X_{n}=1\right)=1 / 2$$

Define for $k \in \mathbb{N}$

$$Y_{k}=\sum_{n=1}^{\infty} X_{k+n-1} / 2^{n}$$

What is the distribution of $Y_{k} ? \quad$ Show that the random variables $Y_{1}$ and $Y_{2}$ are not independent.

Set $S_{n}=Y_{1}+\cdots+Y_{n}$. Show that $S_{n} / n$ converges as $n \rightarrow \infty$ almost surely and determine the limit. [You may use without proof any standard theorem provided you state it clearly.]