State the definitions of a martingale and a stopping time.

State and prove the optional sampling theorem.

If $\left(M_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale, under what conditions is it true that $M_{n}$ converges with probability 1 as $n \rightarrow \infty$ ? Show by an example that some condition is necessary.

A market consists of $K>1$ agents, each of whom is either optimistic or pessimistic. At each time $n=0,1, \ldots$, one of the agents is selected at random, and chooses to talk to one of the other agents (again selected at random), whose type he then adopts. If choices in different periods are independent, show that the proportion of pessimists is a martingale. What can you say about the limiting behaviour of the proportion of pessimists as time $n$ tends to infinity?