Mathematics · Probability Theory
A topic within Probability Theory, itself one of 12 topics in that field and part of Mathematics.
No prior grounding assumed.
What is a Martingale?
Joseph L. Doob · 1971
Explains, for a general mathematical audience, why the martingale property that tomorrow's expected value is today's value is the single idea…
Assumes you know the vocabulary.
Probability with Martingales
David Williams · 1991
Builds measure-theoretic probability around the martingale as the organising idea, arguing that once convergence and optional stopping are…
Primary sources and full treatments.
Regularity Properties of Certain Families of Chance Variables
Joseph L. Doob · 1940
Establishes the upcrossing inequality and the martingale convergence theorem, showing a broad class of stochastic processes converges almost surely…
3 works
Joseph L. Doob
Explains, for a general mathematical audience, why the martingale property that tomorrow's expected value is today's value is the single idea unifying gambling systems, random walks and the convergence theorems built on them.
link checked 17 Sept 2026David Williams
Builds measure-theoretic probability around the martingale as the organising idea, arguing that once convergence and optional stopping are established for martingales, most of the rest of the theory follows as a special case.
Joseph L. Doob
Establishes the upcrossing inequality and the martingale convergence theorem, showing a broad class of stochastic processes converges almost surely from a single combinatorial argument about how often a sequence can cross a band of values.
link checked 17 Sept 2026