TheoremStatement: Literature-sourcedProof: 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 DQM score has mean zero and is unique

Statement

Assume the density-version convention for the path in A dominated differentiable-in-quadratic-mean submodel. If $s \in L^{2} (P)$ is a DQM score, then $P s = 0$ , so $s \in L_{0}^{2} (P)$ (Mean-zero square-integrable functions). Moreover, any two scores satisfying the same DQM expansion agree $P$ -almost surely. The normalization argument itself uses no choice beyond the density versions already supplied in the definition.

Facts & Assumptions

Given: A common-dominated probability path $P_{t}$ through $P = P_{0}$ , with density versions $p_{t}$ and a DQM score $s \in L^{2} (P)$ .

[F1]

DQM means $∥ \sqrt{p_{t}} - \sqrt{p} - (t / 2) s \sqrt{p} ∥_{L^{2} (μ)} = o (t)$ as $t ↓ 0$ (A dominated differentiable-in-quadratic-mean submodel).

[F2]

A Radon–Nikodym density $p_{t}$ represents $P_{t}$ by $P_{t} (E) = \int_{E} p_{t} d μ$ for every measurable $E$ (A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density).

[F3]

The $L^{2}$ pairing is finite and satisfies Cauchy–Schwarz (Cauchy-Schwarz inequality for $L^{2}$ ).

[F4]

The Lebesgue integral is linear on $L^{1} (μ)$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

Proof

technique · Expand the squared norm and use normalization of the path

1.1F1F2

Put $q = \sqrt{p}$ , $q_{t} = \sqrt{p_{t}}$ , and $r_{t} = q_{t} - q - (t / 2) s q$ . By [F1], $∥ r_{t} ∥_{2} = o (t)$ , while [F2] gives $∥ q ∥_{2}^{2} = \int p d μ = 1$ and $∥ q_{t} ∥_{2}^{2} = \int p_{t} d μ = 1$ .

1.2F1algebra

If $s^{'}$ is another score with remainder $r_{t}^{'} = o_{L^{2}} (t)$ , subtracting its expansion from [F1] gives $∥ (t / 2) (s - s^{'}) q ∥_{2} \leq ∥ r_{t} ∥_{2} + ∥ r_{t}^{'} ∥_{2} = o (t)$ .

2.1step 1.1F3F4algebra

Set $u_{t} = (t / 2) s q + r_{t}$ . Since $∥ s q ∥_{2}^{2} = \int s^{2} p d μ = P s^{2} < \infty$ , we have $∥ u_{t} ∥_{2} = O (t)$ and $q_{t} = q + u_{t}$ . The products $q r_{t}$ and $u_{t}^{2}$ are integrable by [F3]. Pointwise expansion and linearity [F4] give $0 = \int (q + u_{t})^{2} d μ - \int q^{2} d μ = t P s + 2 \int q r_{t} d μ + ∥ u_{t} ∥_{2}^{2}$ . By [F3], $∣ \int q r_{t} d μ ∣ \leq ∥ q ∥_{2} ∥ r_{t} ∥_{2} = o (t)$ and $∥ u_{t} ∥_{2}^{2} = O (t^{2}) = o (t)$ . Dividing by $t > 0$ and taking $t ↓ 0$ gives $P s = 0$ .

3.1step 1.2step 2.1algebra∎

Divide by $t / 2$ and let $t ↓ 0$ . Then $∥ (s - s^{'}) q ∥_{2}^{2} = P [(s - s^{'})^{2}] = 0$ , so $s = s^{'}$ as an element of $L^{2} (P)$ . Together with step 2.1 this proves mean zero and uniqueness.

Depends on

Used by

Pathwise differentiability and parameter influence curves Definition
Tangent set of a statistical model at a law Definition
A boundary score cone differs from its tangent space Example
A regular Bernoulli submodel and its score Example
Bounded mean-zero scores generate the unrestricted tangent space Example
A one-sided DQM path yields an iid LAN proxy Lemma

Dependency tree · two levels

25 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

A. W. van der Vaart, Semiparametric Statistics, Lecture 1 §1.2, Lemma 1.8 (standard reference, not scraped)
Edward H. Kennedy, Semiparametric theory, §4 (standard reference, not scraped)