Doob martingale

A martingale formed by taking conditional expectations of a random variable or process with respect to an increasing sequence of information. It tracks how conditional predictions change as information accumulates.

Conditional expectation: A random variable representing the average value of another random variable given specified information. Each martingale value is the conditional expectation of a fixed target given current information.
Integrable random variable: A random variable whose absolute value has finite expectation. Finite expectation ensures the conditional predictions defining the martingale are well-defined.
Azuma–Hoeffding inequality: A concentration inequality bounding deviations of a martingale with bounded increments. Doob martingales let functions of random inputs inherit bounds from changes in individual inputs.
Conditional expectation of a sum of independent variables: A conditional prediction of a sum after some of its independent summands have been revealed. The unrevealed summands retain their means, while each revealed value updates the prediction.
Filtration (probability theory): An increasing sequence of sigma-algebras representing information available over time. The filtration specifies exactly which information each conditional expectation may use.
Conditional independence: Independence between random variables after conditioning on specified information. It can simplify conditional expectations when a process is revealed in stages.
McDiarmid's inequality: A concentration inequality for functions of independent variables with bounded coordinate-wise changes. Its proof exposes inputs one at a time and bounds the resulting Doob-martingale increments.
Pólya's urn: A probability model in which each sampled ball is replaced along with additional balls of the same color. Conditional predictions of the urn's future composition form a martingale under the natural information filtration.
Martingale: A stochastic process whose next value has conditional expectation equal to its present value. Every Doob martingale satisfies this defining no-drift property.
Almost-sure convergence: Convergence of random variables outside an event of probability zero. Martingale convergence theorems often describe the limiting predictions almost surely.

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Azuma's inequalityRelated: It converts a function of independent inputs into a process to which Azuma can be applied.