Alphabeta MathHow 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 is a sequence of real integrable variables such that is -measurable and 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 , and need not be trivial or generated by the differences.
Depends on
Used by
- Square-integrable martingale-difference array and variance clock Definition
- Concentration for adaptively sampled MDP observations Lemma
- Coordinate-wise stochastic approximation converges to a contraction fixed point Lemma
- Martingale differences are orthogonal in l2 Lemma
- Predictably weighted bounded martingale differences have a convergent series Lemma
- Uniform empirical reward and transition confidence bounds Lemma
- Expected-average stationarity under aligned stochastic directions Theorem
- Finite-sample DAgger surrogate-loss guarantee Theorem
- Martingales and martingale differences correspond Theorem
- Nonconvex SGD weighted random-iterate stationarity Theorem
- Private online learning regret under composition Theorem
- Tabular Q-learning converges under per-pair Robbins–Monro steps and full visitation Theorem
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)