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.