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.

Pointwise consistency in probability

Definition

Let $(T_{n})$ be an estimator sequence of an estimand $ψ$ with metric target $(T, d)$ , and fix $θ \in Θ$ . The sequence is consistent in probability at $θ$ if for every $ε > 0$ , $P_{θ}^{\otimes n} ({x \in X^{n} : d (T_{n} (x), ψ (θ)) > ε}) ⟶ 0.$

The error event is measurable: by the triangle inequality, $t \mapsto d (t, ψ (θ))$ is continuous, so its strict superlevel set at $ε$ is open in the target metric topology and hence Borel; its preimage under $T_{n}$ is measurable.

The consistency is pointwise because $θ$ is fixed before the limit. It does not assert uniform convergence over $Θ$ (Estimator sequences in IID experiments).

Depends on

Estimator sequences in IID experiments

Used by

An exact M-estimator can fail without population separation Counterexample
A finite-class M-estimator and its uniform law Example
The Bernoulli sample mean Example
The same Bernoulli law with two different targets Example
Consistency of M-estimators under a uniform law Theorem
Consistency of the IID sample mean under a finite first moment Theorem
Consistency of Z-estimators under a uniform law Theorem
Nonidentifiability precludes simultaneous consistency 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

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 15, PDF pp. 6–7 (standard reference, not scraped)
John Duchi, Theory of Statistics, Stats 300B Lecture 3, Winter 2018, PDF pp. 3–4 (standard reference, not scraped)