DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Martingale difference sequence

Definition

Assume AC The Axiom of Choice for conditional-expectation existence. A martingale difference sequence relative to $(F_{n})_{n \geq 0}$ is a sequence $(D_{n})_{n \geq 1}$ of real integrable variables such that $D_{n}$ is $F_{n}$ -measurable and $E [D_{n} ∣ F_{n - 1}] = 0 a.s. for every n \geq 1.$ Integrability and adaptedness have the timewise meanings of Adapted and integrable stochastic process. Conditional expectations have the class convention of Conditional expectation as an ae class, which is the exact inherited use of AC here. No representative is selected by this definition. There is no $D_{0}$ , and $F_{0}$ need not be trivial or generated by the differences.

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)