The ergodic theorem

Let $T = \left\{ 1,2, \ldots \right\}$ and let $\ell$ be the set of invariant events of $\left(\mathbb {R}^{T}, \mathscr {B}^{T}\right)$. Show that $\ell$ is a $\sigma$-field.

Assume that $X_{1}, X_{2}, \ldots$ is a stationary sequence with autocovariance function $c(m)$. Show that

Assuming that $j^{-1} \sum_{i = 0}^{j-1} c(i) \rightarrow \sigma^{2}$ as $j \rightarrow \infty$, show that

Let $X_{1}, X_{2}, \ldots$ be independent identically distributed random variables with zero mean and unit variance. Let

where the $\alpha_{i}$ are constants satisfying $\sum_{i} \alpha_{i}^{2} < \infty$. Use the martingale convergence theorem to show that the above summation converges almost surely and in mean square. Prove that $n^{-1} \sum_{i = 1}^{n} Y_{i} \rightarrow 0$ a.s. and in mean, as $n \rightarrow \infty$.