State Birkhoff's Almost Everywhere Ergodic Theorem for measure-preserving transformations. Define what it means for a sequence of random variables to be stationary. Explain briefly how the stationarity of a sequence of random variables implies that a particular transformation is measure-preserving.

A bag contains one white ball and one black ball. At each stage of a process one ball is picked from the bag (uniformly at random) and then returned to the bag together with another ball of the same colour. Let $X_{n}$ be a random variable which takes the value 0 if the $n$th ball added to the bag is white and 1 if it is black.

(a) Show that the sequence $X_{1}, X_{2}, X_{3}, \ldots$ is stationary and hence that the proportion of black balls in the bag converges almost surely to some random variable $R$.

(b) Find the distribution of $R$.

[The fact that almost-sure convergence implies convergence in distribution may be used without proof.]