LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

A statistic induces a deterministic probability kernel

Statement

Let $T : (X, A) \to (D, G)$ be a measurable statistic, where $G$ is a sigma-algebra on the action set (Statistics and estimators). Then $K_{T} (x, B) : = δ_{T (x)} (B) = 1_{B} (T (x)), x \in X, B \in G,$ is a probability kernel and therefore a deterministic randomized procedure.

Facts & Assumptions

Given: A measurable statistic $T$ from the data measurable space to the action measurable space.

[F1]

A statistic is a measurable map; hence $T^{- 1} (B) \in A$ for every $B \in G$ . (Statistics and estimators)

[F2]

For each point $d \in D$ , the Dirac set function $δ_{d}$ is a probability measure on $(D, G)$ . (The Dirac set function at a point, A Dirac set function is a probability measure)

[F3]

A probability kernel has probability-measure sections and measurable event evaluations. (Measure kernel and probability kernel)

Proof

technique · direct

1.1givenF2

For each $x \in X$ , the section $B \mapsto K_{T} (x, B) = δ_{T (x)} (B)$ is the probability measure $δ_{T (x)}$ by [F2].

2.1F1F3step 1.1∎

For each $B \in G$ , $x \mapsto K_{T} (x, B) = 1_{B} (T (x)) = 1_{T^{- 1} (B)} (x)$ is $A$ -measurable by [F1]. Thus both clauses of the kernel definition [F3] hold, so $K_{T}$ is a probability kernel, hence a deterministic randomized procedure.

Depends on

Used by

Bias and mean squared error Definition
Diagonal covariance lower bound with bounded-risk saturation Lemma
Gaussian sparse-sequence minimax lower bound Theorem

Dependency tree · two levels

13 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

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.1, PDF pp. 100–102 (standard reference, not scraped)
Peter Hoff, Statistical Decision Problems, §§2–4.2, PDF pp. 2–8 (standard reference, not scraped)