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From Wikipedia

A Doob martingale is a generic construction that is always a martingale. Specifically, consider any set of random variables $$ \vec{X}=X_1, X_2, ..., X_n $$ taking values in a set $A$ for which we are interested in the function $f:A^n \to \Bbb{R}$ and define: $$ B_i=E_{X_{i+1},X_{i+2},...,X_{n}}[f(\vec{X})|X_{1},X_{2},...X_{i}] $$ It is possible to show that $B_i$ is always a martingale regardless of the properties of $X_i$. The sequence $\{B_i\}$ is the Doob martigale for $f$.

  1. I wonder if the function $f:A^n \to \Bbb{R}$ is a given function, so Doob martingale is relative to both $\vec{X}$ and $f$?
  2. What requirement is on $f$?
  3. Can the set of random variables $\{X_1, X_2, ..., X_n\} $ be generalized from finite to infinite (countably or uncountably)?

Thanks!

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    @ Tim : The notation $E_{(X_{i+1,...,X_n)}}$ is really misleading or confusing in my opinion (for example consider such notation in the context of a brownian motion it is really disturbing isn't it). The proof of the fact that $B_i$ is martingale with respect to $\mathcal{F}_i=\sigma(X_1,...,X_i)$ is clear from the proof of lemma 2 (which just the very well known tower property of iterated conditional expectations) in the document of Anupam Gupta given as reference on wiki's page. I shall try to answer your question later on .2012-10-16

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