9.3 Convergence
Theorem 9.3.1 (Martingale convergence theorem). Let \(\{X_n\}\) be a martingale with \(\sup _n E\left |X_n\right | < \infty \). Then there is a random variable \(X\) with \(E\left |X\right |<\infty \) such that \[X_n \underset {a.s.}{\longrightarrow } X .\] If in addition \(\sup _n E\left (X_n^{2}\right )<\infty \), the convergence also holds in quadratic mean and \(E(X) = E(X_0)\).
Note. The two halves of the theorem are the two halves of the remark above. Bounded expectations give almost sure convergence and nothing more — the doubling example satisfies \(E\left |X_n\right |=1\) for every \(n\) and converges almost surely to \(0\) while its mean stays at \(1\). Bounded second moments give quadratic mean convergence as well, and only then does the limit inherit the mean. The doubling example fails that second condition: \(E\left (X_n^{2}\right )=2^{n}\).
This is the cleanest statement of why a fair game can be a certain ruin. The game is fair at every finite stage, and the limit is not fair at all.
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