Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 26J (Probability and Measure)

Kolmogorov zero-one law says that the tail sigma-algebra of independent random variables is trivial: every tail event has probability zero or one. The Birkhoff ergodic theorem says that for a probability-preserving transformation $T$ and integrable $f$, the averages $A_{n}f=n^{-1}{\sum }_{j=0}^{n-1}f\circ T^j$ converge almost surely to $\mathbb{E}(f\mid \mathcal{I})$, where $\mathcal{I}$ is the invariant sigma-algebra. The mean ergodic theorem in $L^p$, $1\le p<\infty$, gives the same convergence in $L^p$ for $f\in L^p$; $p=2$ is the classical von Neumann theorem, and on a probability space the $L^1$ version applies to the integrable case needed here. There is no general supremum-norm convergence assertion for $p=\infty$.

The strong law of large numbers for iid integrable variables is $n^{-1}{\sum }_{j=1}^n{X}_j\to \mathbb{E}{X}_1$ almost surely. To prove it, realize the sequence on its product probability space and take the left shift $T$. The shift preserves the iid product law. Birkhoff applied to $f=X_1$ gives an almost-sure invariant limit $Z$. An invariant event is, modulo null sets, measurable with respect to coordinates after every finite index, hence is a tail event. Kolmogorov's law makes the invariant sigma-algebra trivial, so $Z$ is constant almost surely. Finally the $L^1$ mean ergodic theorem permits passing expectations through the limit; all averages have expectation $\mathbb{E}{X}_1$. This identifies that constant and proves the strong law.