Alphabeta Math
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.

Adapted and integrable stochastic process

Definition

On a filtered probability space Filtration and filtered probability space, a real stochastic process is a sequence (Xn)n∈N0 of real random variables Random elements and real random variables on the same space. It is adapted when Xn−1(B)∈Fn for every n and every real Borel set B. It is integrable when E∣Xn∣<∞ for every n, using Expectation of a nonnegative or integrable random variable.

An integrable adapted process satisfies both conditions at each fixed time. Integrability here does not mean sup⁡nE∣Xn∣<∞: the deterministic process Xn=n has E∣Xn∣=n<∞ at every time but unbounded supremum. Measurability and finiteness of each integral are separate requirements. No versions, limits or choices are selected.

Depends on

Used by

Dependency tree · two levels

9 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