DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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.

Hyperrectangle classes in a Gaussian sequence model

Definition

Write $N_{+} = {1, 2, \dots}$ . Let $ℓ^{2} (N_{+}, R)$ be the real square-summable sequence space from Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ . For $β > 0$ and $M > 0$ , the Gaussian-sequence hyperrectangle of smoothness index $β$ and radius $M$ is $Θ_{β} (M) : = {θ \in ℓ^{2} (N_{+}, R) : \sup_{i \geq 1} i^{1 + 2 β} θ_{i}^{2} \leq M} .$ The definition uses a coordinatewise weighted bound. It is distinct from the Sobolev ellipsoid ${θ : \sum_{i} i^{2 β} θ_{i}^{2} \leq M}$ .

Depends on

Square-summable families on an arbitrary index set and the space $\ell^2(I)$

Used by

Sparse-gap truth causes empirical-Bayes credible-set undercoverage Counterexample
A power-law truth lies in the polished-tail class Example
A sparse-gap truth lies in a hyperrectangle but is undercovered Example

Dependency tree · two levels

17 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

Botond Szabó, Aad van der Vaart and Harry van Zanten, Frequentist coverage of adaptive nonparametric Bayesian credible sets, Annals of Statistics 43(4) (2015), §1.1 (standard reference, not scraped)
B. T. Knapik, A. W. van der Vaart and J. H. van Zanten, Bayesian inverse problems with Gaussian priors, Annals of Statistics 39(5) (2011), Assumption 3.1 (standard reference, not scraped)