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Finite Wick quadratic pairing

Facts & Assumptions

Given: Assume the Axiom of Choice and let $Q \sim N (0, C)$ be a centered scalar Gaussian with variance $C \geq 0$ .

[A1]

The finite-dimensional normal-law suppliers explicitly assume the Axiom of Choice. The Axiom of Choice

[F1]

A centered scalar normal with covariance $C$ is a one-dimensional instance of the multivariate normal law. Multivariate normal law, including singular covariance

[F2]

The finite covariance Wick definition gives $: Q^{2} :_{C} = Q^{2} - C$ . Finite covariance Wick normal powers

[F3]

The Gaussian pairing formula gives $E [Q^{2}] = C$ and $E [Q^{4}] = 3 C^{2}$ . Finite Gaussian normalization and Wick moments

Example

For a finite centered Gaussian coordinate, the expectation of the product of two Wick-subtracted quadratic powers is

$E [(: Q^{2} :_{C}) (: Q^{2} :_{C})] = 2 C^{2} .$

Verification

technique · expand the finite polynomials and apply the fourth-moment pairing formula

Given: $C \geq 0$ and [A1].

1.1F2algebra

By [F2], $(: Q^{2} :_{C}) (: Q^{2} :_{C}) = (Q^{2} - C)^{2} = Q^{4} - 2 C Q^{2} + C^{2}$ .

2.1A1F1F3step 1.1algebra∎

Taking expectations and substituting [F3] gives $3 C^{2} - 2 C \cdot C + C^{2} = 2 C^{2}$ . This also holds at the degenerate endpoint $C = 0$ , where the law is concentrated at zero. The Gaussian law is available under the stated Choice hypothesis [A1].

Finite Wick quadratic pairing

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

Sources