TITLE: Examples of discrete time martingales
QUESTION [5 upvotes]: 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 discrete time martingales?

REPLY [4 votes]: If you're interested in a somewhat off-beat example of a discrete martingale, you could look at 
Rafe Jones, Iterated Galois towers, their associated martingales, and
the $p$-adic Mandelbrot set, Compos. Math. 143 (2007), 1108-1126.
This is really a result in discrete dynamics over finite fields, and Jones uses tools from number theory (Galois theory, chebotarev density theorem) to show that a certain system constructed by iterating polynomials over finite fields is a martingale, and he then uses the convergence of martingales to deduce his final result.
(For those who don't have access to Compositio, there's a pre-print version on Jones's website: http://www.people.carleton.edu/~rfjones/Preprints/Jones_Iterated_Towers_Homepage.pdf)