Intuitively obvious, Wald's theorem is a technical result (with appropriately boring proof) that says that a stopping time behaves "correctly", and can't be used for cheating when gambling.


Let X1, X2, ... be a sequence of random variables with the same distribution, and let T be a stopping time for them, which has finite expectation. Define
S = X1 + ... + XT
(note that the number of terms added to S is itself a random variable; but this is well defined when T is almost surely finite, and we're even assuming finite expectation). Then (the obvious for expected values holds) ES = (ET)(EX1).
When T is not a stopping time, it is easy to "cheat" and obtain very different expectation; see the nonexamples in my stopping time writeup, which all blatantly violate Wald's theorem.

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