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

Example

Take a wavelet basis with $2^{l}$ coefficients at level $l$ and $a_{l} = 2^{l}$ for $l \geq 1$ . The weighted white-noise trace at $δ = 1 / 2$ has level- $l$ contribution

$2^{l} 2^{- l} (\log 2^{l})^{- 1} = \frac{1}{l \log 2},$

so the covariance trace used by this construction diverges. Under countable choice, for every $δ > 1 / 2$ the trace is finite and white noise has a Borel Gaussian law on $H (δ)$ ; under full AC its law is tight. In every infinite-dimensional orthonormal basis, white noise still fails to be an $L^{2}$ -valued random element. This example records the topology and target-law boundary in the weak BvM theorem; it does not assert a separate impossibility theorem for every conceivable $L^{2}$ posterior limit.

Facts & Assumptions

Given: The wavelet-level weights and the standard independent white-noise coordinates.

[F1]

Logarithmic negative Sobolev spaces for white noise: For wavelets the level multiplicity is $2^{l}$ and the weight is $2^{- l} (\log a_{l})^{- 2 δ}$ , with the coarsest finite level treated separately.

[F2]

White noise is a Borel Gaussian law on the negative Sobolev space: Under countable choice, the white-noise series is Borel Gaussian in $H (δ)$ for $δ > 1 / 2$ , while its independent standard-normal coefficients cannot be those of an $L^{2}$ element.

[F3]

The canonical white-noise law is tight under the Axiom of Choice: Under AC, the canonical Borel law on Polish $H (δ)$ is tight.

[A1]

The Axiom of Choice: Full AC is assumed only for the tightness conclusion in [F3]; the trace calculation and independent-coordinate obstruction to an $L^{2}$ realization use no choice axiom.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable choice is assumed for the density-normal calculus interface in [F2]; its Borel Gaussian conclusion is used under this assumption.

Verification

Given: Countable choice, the basis, the weights, and the white-noise coefficient sequence; full AC is additionally assumed for tightness.

1.1F1algebra

At $δ = 1 / 2$ , the $l$ th trace contribution for $l \geq 1$ is $1 / (l \log 2)$ . On each block $2^{m} \leq l < 2^{m + 1}$ its sum is at least $1 / (2 \log 2)$ , so the trace diverges. For $δ > 1 / 2$ the contribution is proportional to $l^{- 2 δ}$ ; on each dyadic block its sum is at most a constant times $2^{m (1 - 2 δ)}$ , a summable geometric sequence. Finite coarsest terms do not affect either conclusion.

1.2A1A2F2F3

For $δ > 1 / 2$ , [F2] under [A2] gives the Borel Gaussian law on $H (δ)$ and the failure of an $L^{2}$ realization; [F3] gives tightness of that law under [A1].

2.1A1A2F2F3step 1.1step 1.2∎

Therefore, under countable choice, the canonical Gaussian target for the weak BvM result is the negative-Sobolev law described in [F2], tight under the additional full-AC branch [F3]; this target-law construction makes no separate claim about posterior convergence in $L^{2}$ or total variation.

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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)