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A finite stack of independent Gaussian vectors is jointly Gaussian

Statement

Assume the Axiom of Choice. Let $m \geq 1$ be finite, let $d_{j} \geq 1$ , and let $Z_{j}$ for $1 \leq j \leq m$ be mutually independent random elements of $R^{d_{j}}$ . Suppose $Z_{j} \sim N_{d_{j}} (a_{j}, Σ_{j})$ , where each $Σ_{j}$ is symmetric positive semidefinite. Put $D = \sum_{j = 1}^{m} d_{j}$ , $Z = (Z_{1}, \dots, Z_{m})$ , $a = (a_{1}, \dots, a_{m})$ , and $Σ = diag (Σ_{1}, \dots, Σ_{m})$ . Then $Z \sim N_{D} (a, Σ) .$ No covariance block needs to be invertible.

Facts & Assumptions

Given: Assume AC and mutually independent Borel random vectors $Z_{j} \sim N_{d_{j}} (a_{j}, Σ_{j})$ for a finite nonempty index set.

[A1]

AC is the explicit axiom used by the multivariate-normal existence, scalar characteristic-function uniqueness, and Cramer–Wold suppliers below. (The Axiom of Choice)

[F1]

Independence of random elements means independence of their generated sigma-algebras. (Independent random elements)

[F2]

A finite mutually independent family of random elements has product joint law. (Independent random elements have product joint law)

[F3]

Under AC, $N_{d} (a, Σ)$ exists for every finite $d \geq 1$ and symmetric positive-semidefinite $Σ$ , and every scalar projection has its specified normal law. (Multivariate normal law, including singular covariance)

[F4]

Under AC, the characteristic function of $N_{d} (a, Σ)$ is $\exp (i u \cdot a - u^{T} Σ u / 2)$ , including singular $Σ$ . (Characteristic function of a multivariate normal law)

[F5]

Characteristic functions of finite sums of mutually independent real random variables multiply. (Characteristic functions under affine maps and independent sums)

[F6]

Under AC, equal characteristic functions determine equal Borel probability laws on $R$ . (Uniqueness of a law from its characteristic function)

[F7]

Under AC, all one-dimensional projection laws determine a Borel probability law on finite-dimensional Euclidean space. (Cramer wold device)

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2F3

By [F2], the finite tuple $Z$ is a Borel random vector with product joint law. For any $u = (u_{1}, \dots, u_{m})$ , each scalar map $z_{j} \mapsto u_{j} \cdot z_{j}$ is Borel, so preimages of Borel sets lie in $σ (Z_{j})$ . Independence [F1] therefore makes the scalars $u_{j} \cdot Z_{j}$ mutually independent. By [A1, F3], each is normal with mean $u_{j} \cdot a_{j}$ and variance $u_{j}^{T} Σ_{j} u_{j}$ .

2.1F4F5F6step 1.1algebra

For any $s \in R$ , [F5] and [F4] give $φ_{u \cdot Z} (s) = \prod_{j = 1}^{m} \exp ⁣ (i s u_{j} \cdot a_{j} - \frac{1}{2} s^{2} u_{j}^{T} Σ_{j} u_{j}) = \exp ⁣ (i s u \cdot a - \frac{1}{2} s^{2} u^{T} Σ u)$ . The last expression is the characteristic function of $N_{1} (u \cdot a, u^{T} Σ u)$ by [F4] in dimension one. Scalar uniqueness [F6] gives $u \cdot Z \sim N_{1} (u \cdot a, u^{T} Σ u)$ , including the zero-variance point-mass case.

3.1A1F3F7step 2.1algebra∎

The block matrix $Σ$ is positive semidefinite because $u^{T} Σ u = \sum_{j} u_{j}^{T} Σ_{j} u_{j} \geq 0$ . By [A1, F3], the law $N_{D} (a, Σ)$ exists and its projection laws are the normal laws found in step 2.1. Cramer–Wold [F7] therefore identifies the law of $Z$ with $N_{D} (a, Σ)$ . This also covers singular blocks and uses no inverse.

Depends on

Used by

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Sources

Nicolai Meinshausen, Time Series Analysis: Notes for the course HS 2022 (standard reference, not scraped)
Dewei Wang, STAT 720 Time Series Analysis, Spring 2015 lecture notes (standard reference, not scraped)