Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector

Let $(X_1,\dots,X_d)$ be a Gaussian random vector on a probability space $(\Omega,\mathcal{F},P)$ and let $\bigl(m,(\mu_i),(a_{ij}),(Z_j)\bigr)$ be any Gaussian representation of it. Then:

1. (Square-integrability) Each $X_i$ is square-integrable.

2. (Moments) With the covariance of square-integrable random variables,

3. (Mean vector and covariance matrix) The tuple $\bigl(\mathbb{E}[X_1],\dots,\mathbb{E}[X_d]\bigr)$ is called the mean vector and the $d\times d$ real matrix $\Sigma$ with entries $\Sigma_{ik}=\operatorname{Cov}(X_i,X_k)$ is called the covariance matrix of $(X_1,\dots,X_d)$. Both are defined directly from $X_1,\dots,X_d$, hence do not depend on the choice of Gaussian representation, and by Claim 2 the displayed formulas hold for every representation. The covariance matrix is symmetric, $\Sigma_{ik}=\Sigma_{ki}$, and positive semidefinite: for all real numbers $c_1,\dots,c_d$,

4. (Existence) There exists a probability space carrying an independent sequence of standard normal random variables, by Existence of Independent Sequences with Prescribed Distributions. On any probability space carrying independent standard normal random variables $Z_1,\dots,Z_m$, and for any real numbers $(\mu_i)$ and $(a_{ij})$, the random variables $X_i=\mu_i+\sum_{j=1}^{m}a_{ij}Z_j$ form a Gaussian random vector admitting $\bigl(m,(\mu_i),(a_{ij}),(Z_j)\bigr)$ as a Gaussian representation.

All sums over $j$ are empty and equal to $0$ when $m=0$, in accordance with Gaussian Random Vectors and Jointly Gaussian Random Variables.