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.

Predictable discrete time process

Definition

On a discrete filtered probability space Filtration and filtered probability space, a real process $(H_{n})_{n \geq 1}$ is predictable if $H_{n}$ is $F_{n - 1}$ -measurable for every $n \geq 1$ . Real means finite-valued, as in Random elements and real random variables. There is no $H_{0}$ convention and predictability alone imposes neither integrability nor boundedness.

Since $F_{n - 1} \subseteq F_{n}$ , every Borel preimage of $H_{n}$ is also in $F_{n}$ . Thus predictability implies adaptedness at the positive times. This is a preimage inclusion, requiring no choice or conditional expectation.

Depends on

Used by

An unbounded predictable transform may lose integrability Counterexample
Compensator and doob decomposition Definition
Discrete martingale transform Definition
Predictable quadratic variation in discrete time Definition

Dependency tree · two levels

6 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)