LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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 common data law forces a two-point tail-loss bound

Statement

Let $(Z, G, R)$ be a probability space and let $Θ$ be a measurable space (Measurable spaces and measurable sets) with $θ_{0}, θ_{1} \in Θ$ . Let $\hat{θ} : Z \to Θ$ be measurable and let $ℓ : Θ \times Θ \to [0, + \infty]$ have measurable sections $a \mapsto ℓ (θ_{i}, a)$ . If $\max {ℓ (θ_{0}, a), ℓ (θ_{1}, a)} \geq κ ℓ (θ_{0}, θ_{1})$ for every $a \in Θ$ , where $κ > 0$ , then $\max_{i = 0, 1} R {ℓ (θ_{i}, \hat{θ}) \geq κ ℓ (θ_{0}, θ_{1})} \geq 1 / 2$ .

Facts & Assumptions

Given: A probability space $(Z, G, R)$ , two parameter values, a measurable estimator, measurable loss sections at those values, and the stated pairwise loss-separation condition.

[F1]

Probability measures and probability spaces: $R (Z) = 1$ .

[F2]

Measurable spaces and measurable sets: $Θ$ has a specified sigma-algebra.

Proof

1.1givenF2choose

For every $z \in Z$ , apply the separation condition to $a = \hat{θ} (z)$ . At least one of $ℓ (θ_{0}, \hat{θ} (z))$ and $ℓ (θ_{1}, \hat{θ} (z))$ is at least $κ ℓ (θ_{0}, θ_{1})$ . The loss-event sets are measurable by [given] and [F2], so the two event indicators sum to at least one pointwise.

2.1F1F2step 1.1algebra∎

Integrating that indicator inequality under the probability measure $R$ gives $R (E_{0}) + R (E_{1}) \geq 1$ , where $E_{i} = {ℓ (θ_{i}, \hat{θ}) \geq κ ℓ (θ_{0}, θ_{1})}$ . Hence at least one probability is at least $1 / 2$ .

Depends on

Used by

Huber contamination modulus lower bound for tail risk Theorem

Dependency tree · two levels

5 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

Philippe Rigollet, 18.S997: High-Dimensional Statistics, Lecture Notes, Chapter 5, §5.3 (standard reference, not scraped)