LemmaStatement: AI-adaptedProof: AI-generatedPipeline-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.

Second-order kernel convolution has an L² bias bound

Statement

Assume ${A C}_{ω}$ . Let $f$ be a density version in $W^{2, 2} (R^{d})$ , and let $K$ be a normalized Borel kernel with $K \in L^{1}$ , $\int u_{j} K (u) d u = 0$ for every $j$ , and $M_{2} (K) : = \int_{R^{d}} ∣ K (u) ∣ ∥ u ∥^{2} d u < \infty .$ Define the convolution $S_{h} f = \int K (u) f (\cdot - h u) d u$ as an $L^{2}$ equivalence class. With $∥ D^{2} f ∥_{2, F}^{2} : = \sum_{j, k = 1}^{d} ∥ \partial_{j k} f ∥_{L^{2}}^{2},$ one has, for every $h > 0$ , $∥ S_{h} f - f ∥_{L^{2}} \leq \frac{h^{2}}{2} M_{2} (K) ∥ D^{2} f ∥_{2, F} .$

Facts & Assumptions

Given: ${A C}_{ω}$ , $d \geq 1$ , a density version $f \in W^{2, 2} (R^{d})$ , and a normalized Borel kernel $K$ with the displayed moment conditions.

[A1]

The Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Under ${A C}_{ω}$ , $W^{2, 2}$ consists of $L^{2}$ classes whose weak partial derivatives through order two are in $L^{2}$ (Integer-order Sobolev spaces and their norms).

[F2]

Under ${A C}_{ω}$ , $C_{c}^{\infty} (R^{d})$ is dense in $W^{2, 2} (R^{d})$ : for every $ϵ > 0$ there is a smooth compactly supported $g$ with $∥ g - f ∥_{W^{2, 2}} < ϵ$ (Compactly supported smooth functions are dense in W^{k,p}(R^n)).

[F3]

Tonelli interchanges the integrals of nonnegative product-measurable functions on sigma-finite product spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, The product measure of two sigma-finite measure spaces, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique).

[F4]

Under ${A C}_{ω}$ , a $C^{1}$ diffeomorphism satisfies the change-of-variables formula for integrable functions; translations therefore preserve $L^{2}$ norms (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F5]

Holder's inequality with exponents $2, 2$ gives $∣ \int a b ∣^{2} \leq (\int ∣ a ∣^{2}) (\int ∣ b ∣^{2})$ when the square integrals are finite (Holder's inequality for integrals, including the endpoint cases).

[F6]

The total-derivative chain rule differentiates a smooth function along an affine line (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F7]

Integration by parts and the second fundamental theorem evaluate $\int_{0}^{1} (1 - t) q^{''} (t) d t = q (1) - q (0) - q^{'} (0)$ for $q \in C^{2} ([0, 1])$ (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

Proof

1.1givenA1F3F4F5F6F7algebra

First take $g \in C_{c}^{\infty} (R^{d})$ . For each $x, u$ , set $q (t) = g (x - t h u)$ . The chain rule [F6] gives $q^{'} (0) = - h \nabla g (x) \cdot u$ and $q^{''} (t) = h^{2} u^{T} D^{2} g (x - t h u) u$ . Integration by parts and the second fundamental theorem [F7] give $g (x - h u) = g (x) - h \nabla g (x) \cdot u + h^{2} \int_{0}^{1} (1 - t) u^{T} D^{2} g (x - t h u) u d t .$ The first moments are absolutely integrable since $∣ u_{j} ∣ \leq 1 + ∥ u ∥^{2}$ and $K \in L^{1}$ with $M_{2} (K) < \infty$ . Integrating against $K (u)$ therefore cancels the constant and linear terms by $\int K = 1$ and $\int u_{j} K (u) d u = 0$ . Euclidean Cauchy–Schwarz gives $∣ u^{T} D^{2} g (y) u ∣ \leq ∥ u ∥ ∥ D^{2} g (y) u ∥ \leq ∥ u ∥^{2} ∥ D^{2} g (y) ∥_{F}$ . Apply Holder [F5] with the finite measure $∣ K (u) ∣ (1 - t) ∥ u ∥^{2} d u d t$ , whose total mass is $M_{2} (K) / 2$ , to obtain $∣ S_{h} g (x) - g (x) ∣^{2} \leq h^{4} (\frac{M_{2} (K)}{2}) \int_{R^{d}} ⁣ \int_{0}^{1} ∣ K (u) ∣ (1 - t) ∥ u ∥^{2} ∥ D^{2} g (x - t h u) ∥_{F}^{2} d t d u .$ Tonelli [F3] and translation invariance from [F4] then give $∥ S_{h} g - g ∥_{2}^{2} \leq h^{4} {(\frac{M_{2} (K)}{2})}^{2} ∥ D^{2} g ∥_{2, F}^{2} .$

2.1givenA1F1F2F3F4F5step 1.1algebra∎

For every Borel representative $q \in L^{2}$ , the integrand $(x, u) \mapsto K (u) q (x - h u)$ is product-measurable. Holder [F5] and Tonelli [F3] show its defining integral exists absolutely for almost every $x$ and $∥ S_{h} q ∥_{2}^{2} \leq ∥ K ∥_{1} \int_{R^{d}} ∣ K (u) ∣ \int_{R^{d}} ∣ q (x - h u) ∣^{2} d x d u = ∥ K ∥_{1}^{2} ∥ q ∥_{2}^{2},$ by Holder [F5], Tonelli [F3], and translation invariance [F4]. If two Borel representatives agree almost everywhere, translation invariance makes their translates agree almost everywhere for each $u$ ; Tonelli [F3] then makes their convolution integrals agree for almost every $x$ . Thus $S_{h}$ is well-defined as an $L^{2}$ equivalence class on the classes used here. Given $ϵ > 0$ , choose $g \in C_{c}^{\infty}$ as in [F2]. The finite sum defining $∥ D^{2} (g - f) ∥_{2, F}$ is at most $d ∥ g - f ∥_{W^{2, 2}}$ , so step 1.1 and the operator bound just proved imply $∥ S_{h} f - f ∥_{2} \leq (∥ K ∥_{1} + 1) ϵ + \frac{h^{2}}{2} M_{2} (K) (∥ D^{2} f ∥_{2, F} + d ϵ) .$ Letting $ϵ ↓ 0$ proves the asserted bound.

Depends on

Used by

Global MISE upper bound for the KDE Theorem

Dependency tree · two levels

94 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

Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT OpenCourseWare 14.385, Fall 2007, integrated MSE calculation, PDF pp. 5–7 (standard reference, not scraped)
Adrian Bowman and Ludger Evers, Nonparametric Smoothing, APTS 2013–14 course notes, §1.2.1, PDF p. 8 (standard reference, not scraped)