ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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 ratio transformation

Example

Give $Ω = {- 1, 1}^{2}$ equal mass $1 / 4$ at each point, and let $Z = (Z_{1}, Z_{2})$ be the coordinate vector. Put $θ = (1, 2)$ , $T_{n} = θ + n^{- 1 / 2} Z$ for $n \geq 1$ , and define $g (x, y) = x / y$ for $y \neq 0$ and $g (x, 0) = 0$ . The denominator of $g (T_{n})$ is positive for every $n$ . The first-order delta method gives $\sqrt{n} (\frac{T_{n, 1}}{T_{n, 2}} - \frac{1}{2}) \Rightarrow \frac{1}{2} Z_{1} - \frac{1}{4} Z_{2} .$ The limit takes the values $1 / 4, 3 / 4, - 3 / 4, - 1 / 4$ with equal probability and has variance $5 / 16$ .

Facts & Assumptions

Given: The four-point probability law and sequence in the example.

[A1]

$Ω = {- 1, 1}^{2}$ has full sigma-field and assigns mass $1 / 4$ to each point; the coordinate variables $Z_{1}, Z_{2}$ are independent Rademacher variables.

[F1]

First- and second-order delta method for random vectors: if $r_{n} > 0$ is deterministic with $r_{n} \to \infty$ , $r_{n} (T_{n} - θ) \Rightarrow Z$ , and the Borel map $g$ has derivative $A$ at $θ$ with first-order remainder $o (∥ h ∥)$ , then $r_{n} (g (T_{n}) - g (θ)) \Rightarrow A Z$ .

Verification

Given: The finite four-point law, $r_{n} = \sqrt{n}$ , $θ = (1, 2)$ , and the ratio map $g$ in the example.

1.1A1givenalgebra

By [A1], $Z_{2} \in {- 1, 1}$ , so $T_{n, 2} = 2 + Z_{2} / \sqrt{n} \geq 1$ for every $n \geq 1$ . For $∣ v ∣ < 1$ , the exact remainder after subtracting the linear map is $g (1 + u, 2 + v) - \frac{1}{2} - (u / 2 - v / 4) = - (2 u - v) v / [2 (4 + 2 v)]$ , whose absolute value is bounded by a constant times $u^{2} + v^{2}$ . Thus the derivative at $(1, 2)$ is $(u, v) \mapsto u / 2 - v / 4$ . Also $\sqrt{n} (T_{n} - θ) = Z$ exactly.

2.1F1step 1.1given

Apply [F1] with this derivative to obtain the stated limit.

3.1A1givenstep 2.1algebra∎

By [A1], the four equiprobable values of $\frac{1}{2} Z_{1} - \frac{1}{4} Z_{2}$ are $1 / 4, 3 / 4, - 3 / 4, - 1 / 4$ . Their mean is zero and their second moment is $(1 + 9 + 9 + 1) / 64 = 5 / 16$ , which is the stated variance.

Depends on

First- and second-order delta method for random vectors

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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.