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.
Adapted and integrable stochastic process
Definition
On a filtered probability space Filtration and filtered probability space, a real stochastic process is a sequence of real random variables Random elements and real random variables on the same space. It is adapted when for every and every real Borel set . It is integrable when for every , 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 : the deterministic process has 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
- Empirical-Bernstein confidence-sequence interval update Algorithm
- Minibatch stochastic gradient descent Algorithm
- Stochastic gradient descent Algorithm
- Compensator and doob decomposition Definition
- Conditionally sub-Gaussian increments Definition
- Confidence sequence for a time-varying target Definition
- Discrete martingale transform Definition
- Filtration-adapted stochastic-gradient oracles and Robbins–Monro updates Definition
- Martingale difference sequence Definition
- Martingale submartingale and supermartingale Definition
- Natural filtration of a process Definition
- One-sided sub-exponential uniform boundary Definition
- Stopped random variable and stopped process Definition
- Constant-step SGD can have a nonzero noise floor Example
- Uniform empirical-risk sampling gives an unbiased bounded-variance gradient Example
- First hitting time of an adapted process is a stopping time Lemma
- Robbins–Siegmund almost-supermartingale convergence Lemma
- Empirical-Bernstein confidence sequence under adaptive sampling Theorem
- Expected-average stationarity under aligned stochastic directions Theorem
- Finite-sample DAgger surrogate-loss guarantee Theorem
- Nonconvex SGD weighted random-iterate stationarity Theorem
- Terminal error bounds for a sequential likelihood-ratio test Theorem
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
- van der Vaart, Martingales, Diffusions and Financial Mathematics (standard reference, not scraped)