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 $f$.
- 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 $f$?
- What requirement is on $f$?
- Can the set of random variables $\{X_1, X_2, ..., X_n\} $ be generalized from finite to infinite (countably or uncountably)?
Thanks!