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 logarithmic transformation

Example

Let $Ω = {- 1, 1}$ with each point having probability $1 / 2$ , let $Z (ω) = ω$ , and set $T_{n} = \max {1 / 2, 1 + Z / \sqrt{n}}$ for $n \geq 1$ . Define the Borel map $g (t) = \log t$ for $t > 0$ and $g (t) = 0$ for $t \leq 0$ . Then $\sqrt{n} {\log T_{n} - \log 1} \Rightarrow Z .$ Thus the log transformation has first-order scale $\sqrt{n}$ and derivative $g^{'} (1) = 1$ in this example.

Facts & Assumptions

Given: The finite two-point law and sequence in the example.

[A1]

$Ω = {- 1, 1}$ has full sigma-field and assigns mass $1 / 2$ to each point; $Z (ω) = ω$ .

[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 two-point law, $r_{n} = \sqrt{n}$ , $θ = 1$ , and the function $g$ in the example.

1.1A1givenalgebra

By [A1], $Z \in {- 1, 1}$ . For $n \geq 4$ , $1 + Z / \sqrt{n} \geq 1 - 1 / 2 = 1 / 2$ , so $T_{n} = 1 + Z / \sqrt{n}$ and $\sqrt{n} (T_{n} - 1) = Z$ exactly. Hence $\sqrt{n} (T_{n} - 1) \Rightarrow Z$ .

2.1F1givenstep 1.1∎

The map $g$ is differentiable at $1$ with derivative $1$ . Apply [F1] to obtain $\sqrt{n} {\log T_{n} - \log 1} \Rightarrow Z$ .

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

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