DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Gaussian-mixture confidence-sequence boundary

Definition

Let $σ^{2} > 0$ be a supplied scale, let $ρ > 0$ be a tuning parameter, and let $α \in (0, 1)$ . For $v \geq 0$ define $u_{α} (v) : = \sqrt{(v + ρ) \log ⁣ (\frac{v + ρ}{α^{2} ρ})} .$ The expression is real because $(v + ρ) / (α^{2} ρ) > 1$ . For a real data sequence $x_{1}, x_{2}, \dots$ , set ${\overset{ˉ}{x}}_{t} = t^{- 1} \sum_{i = 1}^{t} x_{i}$ and define $C_{0} : = R, C_{t} : = [{\overset{ˉ}{x}}_{t} - \frac{u_{α} (t σ^{2})}{t}, {\overset{ˉ}{x}}_{t} + \frac{u_{α} (t σ^{2})}{t}] (t \geq 1) .$ The endpoints are included. This definition specifies an interval rule only; it makes no coverage or time-uniform validity assertion.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon, Time-uniform, nonparametric, nonasymptotic confidence sequences, Proposition 5 (standard reference, not scraped)
Aaditya Ramdas, Martingales 1: Concentration Inequalities — Conjugate Mixtures, 36-771 lecture notes, Carnegie Mellon University, Fall 2018, Proposition 1 (standard reference, not scraped)