TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge 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.

Asymptotic linearity implies asymptotic normality

Statement

Assume the Axiom of Choice. Let $X_{1}, X_{2}, \dots$ be iid with law $P$ , let $k < \infty$ , and let ${\hat{ψ}}_{n}$ estimate $ψ (P) \in R^{k}$ . If it is asymptotically linear at $P$ with influence function $φ \in L_{0}^{2} (P)^{k}$ , then

$\sqrt{n} {{\hat{ψ}}_{n} - ψ (P)} \Rightarrow N_{k} (0, Σ_{φ}), Σ_{φ} = P [φ φ^{T}] .$

The covariance may be singular. This is a distributional limit only; the theorem asserts neither local regularity nor estimator attainment of an efficiency bound.

Facts & Assumptions

Given: The iid experiment, the finite-dimensional estimator and target, asymptotic linearity with $φ \in L_{0}^{2} (P)^{k}$ , and AC.

[F1]

Asymptotically linear estimators and influence functions: ${\hat{ψ}}_{n} - ψ (P) = (P_{n} - P) φ + o_{P} (n^{- 1 / 2})$ , where $P φ = 0$ and $P ∥ φ ∥^{2} < \infty$ .

[F2]

Multivariate iid central limit theorem: under AC, normalized sums of iid finite-second-moment vectors converge to the centered multivariate normal with their covariance, including singular covariance.

[F3]

Continuous mapping theorem: continuous linear maps preserve convergence in distribution.

[F4]

Slutsky's theorem for real random variables: adding a real random variable converging to zero in probability preserves a real distributional limit.

[F5]

Cramer wold device: convergence of every one-dimensional projection to the corresponding projection of a specified Borel probability law implies vector convergence in distribution; this use assumes AC.

[A1]

The Axiom of Choice: AC is assumed by [F2] and [F5].

Proof

Given: AC, iid $X_{i} \sim P$ , $k < \infty$ , and an estimator satisfying [F1].

1.1A1F1F2givenalgebra

Put $Z_{n} = n^{- 1 / 2} \sum_{i = 1}^{n} φ (X_{i})$ and $R_{n} = \sqrt{n} {{\hat{ψ}}_{n} - ψ (P)} - Z_{n}$ . By [F1], $R_{n} \to 0$ in probability. The vectors $φ (X_{i})$ are iid, centered, and have finite second moment, so [F2] gives $Z_{n} \Rightarrow Z \sim N_{k} (0, Σ_{φ})$ . The covariance is positive semidefinite by $a^{T} Σ_{φ} a = P [(a^{T} φ)^{2}] \geq 0$ for every $a \in R^{k}$ .

2.1F3F4step 1.1given

Fix any $a \in R^{k}$ . By [F3], $a^{T} Z_{n} \Rightarrow a^{T} Z$ . Also $∣ a^{T} R_{n} ∣ \leq ∥ a ∥ ∥ R_{n} ∥$ , so $a^{T} R_{n} \to 0$ in probability. Apply [F4] to obtain $a^{T} (Z_{n} + R_{n}) \Rightarrow a^{T} Z$ .

3.1A1F5step 1.1step 2.1∎

Step 2.1 holds for every $a$ , and $Z$ is the specified Borel Gaussian law from step 1.1. [F5] therefore gives $Z_{n} + R_{n} \Rightarrow Z$ , which is the asserted limit. No nonsingularity of $Σ_{φ}$ is used.

Depends on

Used by

Dependency tree · two levels

35 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 Notes in Mathematics 1781, Lecture 2, §2.9 (Springer, 2002) (standard reference, not scraped)
Edward H. Kennedy, Semiparametric theory, §§3–4 (2017) (standard reference, not scraped)