LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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The KDE is jointly measurable in sample and evaluation point

Statement

For a Borel normalized kernel $K$ , $h > 0$ and $n \geq 1$ , the map $Q_{h} : (R^{d})^{n} \times R^{d} \to R, (z_{1}, \dots, z_{n}, x) \mapsto \frac{1}{n h^{d}} \sum_{i = 1}^{n} K ⁣ (\frac{x - z_{i}}{h})$ is Borel measurable for the finite product Borel sigma-algebra.

Facts & Assumptions

Given: A Borel normalized kernel $K$ , $h > 0$ , $n \geq 1$ , and the finite product Borel data and evaluation spaces.

[F1]

The sample space carries the finite product Borel sigma-algebra, and $K$ is Borel (Density targets and the kernel density estimator, The product sigma-algebra and its finite iterates).

[F2]

Composition of measurable maps is measurable (Composition with a Borel measurable outer map preserves measurability).

[F3]

Scalar multiples and finite sums of real measurable functions are measurable (Arithmetic and lattice operations preserve measurability whenever they are defined).

[F4]

For positive integers $a, b$ , $B (R^{a}) \otimes B (R^{b}) = B (R^{a + b})$ under the coordinate identification (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}).

Proof

1.1givenF1F2F4

Induct on the number of factors using the finite product recursion in [F1] and the two-factor equality [F4]. Starting with $B (R^{d})$ , this identifies the specified $(n + 1)$ -factor sigma-algebra with $B (R^{(n + 1) d})$ . For each $i$ , the affine map $(z_{1}, \dots, z_{n}, x) \mapsto (x - z_{i}) / h$ is continuous on this Euclidean space: inverse images of open sets are open, hence inverse images of Borel sets are Borel by closure under complements and countable unions. It is therefore measurable for the specified product sigma-algebra. Composing with the Borel map $K$ gives a measurable real function by [F2].

2.1step 1.1F3∎

Applying [F3] to the finite sum of these functions and multiplying by the constant $(n h^{d})^{- 1}$ proves that $Q_{h}$ is Borel measurable.

Depends on

Used by

Global MISE upper bound for the KDE Theorem

Dependency tree · two levels

34 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, PDF pp. 1–3 (standard reference, not scraped)
Adrian Bowman and Ludger Evers, Nonparametric Smoothing, APTS 2013–14 course notes, §1.2, PDF pp. 6–7 (standard reference, not scraped)