The mathematics of uncertainty and random structure.
12 topics · 23 curated works
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…
+3 more at this level
Assumes you know the vocabulary.
Dynamical Theories of Brownian Motion
Edward Nelson · 1967
Argues Brownian motion can be derived from an underlying stochastic mechanics rather than merely postulated, reconstructing it from a Newtonian model…
+5 more at this level
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…
+12 more at this level
12 of 23 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 2026Peter G. Doyle & J. Laurie Snell
Shows that the trajectory of a random walk on a graph and the flow of current through the same graph as a resistor network are the same mathematics seen twice, using each to build intuition for the other.
link checked 17 Sept 2026Charles M. Grinstead & J. Laurie Snell
Builds probability from discrete counting arguments up through Markov chains and Brownian motion using worked simulations throughout, distributed freely as a replacement for a standard undergraduate course text.
link checked 17 Sept 2026Hannah Fry (The Royal Institution)
Shows that luck is mostly mis-read probability, and that people systematically misjudge independent events, coincidence and small samples in ways that can be demonstrated live.
link checked 17 Sept 2026Edward Nelson
Argues Brownian motion can be derived from an underlying stochastic mechanics rather than merely postulated, reconstructing it from a Newtonian model of a particle buffeted by random impacts.
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.
Richard Durrett
Teaches Markov chains, Poisson processes and martingales through applied examples chosen for their own sake, arguing intuition for a stochastic process should come before its general theory.
link checked 17 Sept 2026David Aldous & James Allen Fill
Collects the spectral, coupling and electrical-network methods for bounding how fast a Markov chain converges to equilibrium, treating a graph's random walk and its resistor-network analogue as one problem throughout.
link checked 17 Sept 2026Geoffrey Grimmett
Argues percolation, the Ising model and random graphs are variations on one question — how local randomness on a lattice produces global connectivity — and develops them together rather than as separate subjects.
link checked 17 Sept 2026Roman Vershynin
Collects the concentration inequalities, covering numbers and random matrix bounds that make high-dimensional probability usable in data science, arguing most of the field reduces to controlling how far a random quantity can stray from its mean.
link checked 17 Sept 2026Joseph 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 2026Kiyosi Itô
Defines an integral against Brownian motion's own erratic path, founding stochastic calculus on the claim that such an integral can be made rigorous despite the path having unbounded variation.
link checked 17 Sept 2026