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Uncorrelated jointly Gaussian Banach random elements are independent

Statement

Assume AC and countable choice. Let $X$ and $Y$ be jointly centered Gaussian Borel random elements taking values in real separable Banach spaces $B_{1}$ and $B_{2}$ . Here jointly Gaussian means every finite family of continuous linear coordinates of $(X, Y)$ has a multivariate normal law. If $E [ℓ (X) m (Y)] = 0$ for every $ℓ \in B_{1}^{*}$ and $m \in B_{2}^{*}$ , then $X$ and $Y$ are independent.

Facts & Assumptions

[A1]

AC and countable choice are hypotheses of the Statement (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). AC meets the finite Gaussian interfaces' choice hypothesis; countable choice and HB from [A2] meet the countable norming-family interface. The product-measure interface requires sigma-finiteness, which holds because both marginal laws are probabilities.

[A2]

Under AC, the real dominated-extension theorem supplies the Hahn–Banach principle HB used by the dual-norming interface (Hahn-Banach dominated extension theorem for real vector spaces, The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

A centered multivariate normal vector has characteristic function $u \mapsto e^{- u^{T} Σ u / 2}$ , which determines its law (Multivariate normal law, including singular covariance, Characteristic function of a multivariate normal law).

[F2]

Countable norming coordinates determine a Borel probability law on a separable Banach space (A separable Banach space has a countable norming family, A countable norming family determines Borel laws).

[F3]

The product probability of two Borel probabilities on these Polish spaces exists and has the product values on measurable rectangles (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique).

[F4]

Measures agreeing on a generating pi-system containing the whole space, with equal finite total masses, agree on the generated sigma-algebra (Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system). Here the joint law and product law both assign mass one to $B_{1} \times B_{2}$ .

[F5]

Independence is equivalent to the product identity on all measurable rectangles (Independent random elements, Independent random elements are characterized by finite rectangle probabilities).

Proof

Given: AC, countable choice, jointly centered Gaussian Borel random elements $X, Y$ with values in $B_{1}, B_{2}$ , and zero mixed covariance for every pair of continuous linear functionals.

1.1A1A2F1F2givenchoose

Choose countable norming families $(ℓ_{i})$ for $B_{1}$ and $(m_{j})$ for $B_{2}$ . For any finite subfamilies, the joint coordinate vector is centered multivariate normal. Its covariance matrix has zero off-diagonal block by the hypothesis, so the characteristic function in [F1] is the product of the two marginal characteristic functions. The product law has that same characteristic function, so uniqueness in [F1] proves the finite coordinate vectors are independent.

2.1A1A2F2F3F4F5step 1.1algebra∎

Finite-coordinate cylinder rectangles form a pi-system generating $B (B_{1}) \otimes B (B_{2})$ by [F2]. The joint law of $(X, Y)$ and the product of its marginal laws agree on every such rectangle by step 1.1; [F3] supplies that product measure. The cylinder pi-system contains $B_{1} \times B_{2}$ , and both laws have mass one, so the constant exhaustion meets [F4] and gives equality on the generated sigma-algebra. The rectangle criterion [F5] now gives independence of $X$ and $Y$ .

Depends on

Used by

Fernique integrability for a separable Banach Gaussian law Theorem

Dependency tree · two levels

62 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

A. Lunardi, M. Miranda, and D. Pallara, Infinite Dimensional Analysis (standard reference, not scraped)
M. Hairer, An Introduction to Stochastic PDEs (standard reference, not scraped)