In probability, a martingale is given by a sequence of integrable random variables (S_n) and an increasing sequence of $\sigma$-algebras ${\cal F}_n$ such that $S_n$ is ${\cal F}_n$-measurable and $E(S_{n+1} \mid {\cal F}_n) = S_{n}$.

This is an important notion because there are many results concerning convergence of martingales sequences, e.g. if it is bounded in $L^2$ then it converges in $L^2$ norm and $a.e.$

If $X_i$ is a sequence of i.i.d. random variables and ${\cal F}_n = \sigma(X_i, i\leq n)$, then the following sequences are martingales:

• $S_n - E(S_n)$,

• $exp(S_n)/E(exp(S_n))$,

• $(S_n)^2-E(S_n^2)$,

These are used in the theory of random walks to compute e.g. the mean time before reaching a given state.

Are there any other interesting examples of martingales?