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The canonical white-noise law is tight under the Axiom of Choice

Statement

Assume AC. For $δ > 1 / 2$ , the canonical white-noise law $N$ on the Polish space $H (δ)$ from White noise is a Borel Gaussian law on the negative Sobolev space is tight.

Facts & Assumptions

[F1]

White noise is a Borel Gaussian law on the negative Sobolev space: The white-noise coefficient series defines a Borel probability law $N$ on $H (δ)$ for $δ > 1 / 2$ .

[F2]

Prokhorov tightness theorem on polish spaces: Assume AC. A family of Borel probabilities on a Polish space is tight if and only if it is relatively sequentially compact for weak convergence.

[F3]

Polish spaces are separable completely metrizable spaces: A Polish space is separable and completely metrizable.

[F4]

Logarithmic negative Sobolev spaces for white noise: $H (δ)$ is separable and complete with its weighted coefficient norm.

[A1]

The Axiom of Choice: AC is used exactly in applying Prokhorov's theorem in step 2.1.

Given: AC, $δ > 1 / 2$ , and the Borel probability $N$ on $H (δ)$ .

Proof

1.1F1F3F4given

By [F4], $H (δ)$ is separable and complete; the definition of Polish space [F3] therefore makes it Polish.

2.1A1F2givenstep 1.1∎

Every sequence in the singleton family ${N}$ is constant and therefore converges weakly to $N$ ; thus ${N}$ is relatively sequentially compact. Applying [F2] under [A1] proves that ${N}$ , and hence the law $N$ , is tight.

Depends on

Used by

White noise has a negative-Sobolev law but no L2-valued law Example

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Sources

Castillo and Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise, §1.2, printed pp. 2005–2006 (standard reference, not scraped)
Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §7.3.4, printed pp. 594–595 (standard reference, not scraped)