the martingale convergence theorem
A fair game's fortune jitters up and down without any built-in pull toward a fixed value, so it is surprising that, as long as the swings cannot grow without bound on average, the fortune must eventually settle down to a definite (random) limit. That is the content of the martingale convergence theorem, one of the deepest and most useful results in probability: a martingale that stays bounded in an average-size sense converges almost surely. Endless oscillation is impossible for a process that is, on average, leashed.
The precise statement: if M_n is a martingale (or a submartingale, or a supermartingale) that is L^1-bounded, meaning sup over n of E[|M_n|] is finite, then there is a random variable M_infinity with E[|M_infinity|] finite such that M_n converges to M_infinity almost surely (for almost every outcome, the path has a genuine limit). A common, easy-to-check sufficient condition is that the martingale is bounded above or below by a constant, or non-negative — a non-negative supermartingale always converges. The proof runs through the upcrossing inequality: L^1-boundedness forces the expected number of crossings of every gap [a, b] to be finite, so the path cannot oscillate across any band infinitely often, and a non-oscillating bounded-in-average sequence must converge.
The single most important caveat: almost-sure convergence does NOT imply convergence in mean, and the limit's expectation can disagree with the martingale's. The textbook counterexample is the doubling-down fortune (or 'exponential martingale' built from a product of mean-one variables): it is a non-negative martingale with E[M_n] = 1 for all n, yet it converges almost surely to M_infinity = 0, so E[M_infinity] = 0 not 1. Mass escaped to infinity along rare paths. To also get convergence in L^1 — and the clean identity M_n = E[M_infinity given F_n] — you need the extra hypothesis of uniform integrability. The bare theorem gives you a limit; uniform integrability is what makes that limit behave.
A non-negative martingale always converges, since it is automatically L^1-bounded (E[|M_n|] = E[M_n] = E[M_0]). But beware: the product martingale Y_n = X_1 * ... * X_n with each X_i in {0, 2} of mean 1 has E[Y_n] = 1 for all n, yet Y_n converges almost surely to 0 (a single zero factor kills it forever). The limit exists but E[Y_infinity] = 0 differs from E[Y_n] = 1.
L^1-bounded martingales converge almost surely — but the limit's mean need not match unless you add uniform integrability.
Almost-sure convergence does NOT give convergence in mean: a non-negative martingale with E[M_n] = 1 can converge to a limit whose mean is 0. Uniform integrability is what closes that gap.