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

Step 0 (almost surely equal random vectors). Let $U\in\mathcal{L}^{2}(P;\mathbb{R}^{d})$ and let $X$ be a random vector in $\mathbb{R}^{d}$ with $X=U$ almost surely. We show that $X\in\mathcal{L}^{2}(P;\mathbb{R}^{d})$, $\mathbb{E}X=\mathbb{E}U$ and $\operatorname{Cov}(X)=\operatorname{Cov}(U)$.

Since $|X|^{2}=|U|^{2}$ almost surely, The Lebesgue Integral and Null Sets (ii) gives $\int|X|^{2}\,dP=\int|U|^{2}\,dP<\infty$, so $X\in\mathcal{L}^{2}(P;\mathbb{R}^{d})$; in terms of the inner product space $L^{2}(P;\mathbb{R}^{d})$ of L²(μ;ℝⁿ) Is a Real Hilbert Space (i), $[X]$ and $[U]$ have the same norm by L²(μ;ℝⁿ) Is a Real Hilbert Space (ii). By Rules for Expectations, Variances and Covariances (xi), $U$ and $X$ are integrable. Each coordinate $X_{i}$ agrees almost surely with the integrable $U_{i}$, so $\mathbb{E}X_{i}=\mathbb{E}U_{i}$ by The Lebesgue Integral and Null Sets (iii), that is, $\mathbb{E}X=\mathbb{E}U$. As the means agree, every entry of the centred product $(X-\mathbb{E}X)(X-\mathbb{E}X)^{\top}$ agrees almost surely with the corresponding entry of $(U-\mathbb{E}U)(U-\mathbb{E}U)^{\top}$, which is integrable by Rules for Expectations, Variances and Covariances (xi); so the entries have the same expectations, again by The Lebesgue Integral and Null Sets (iii). With Rules for Expectations, Variances and Covariances (xi),

$$\operatorname{Cov}(X)=\mathbb{E}\bigl[(X-\mathbb{E}X)(X-\mathbb{E}X)^{\top}\bigr]=\mathbb{E}\bigl[(U-\mathbb{E}U)(U-\mathbb{E}U)^{\top}\bigr]=\operatorname{Cov}(U).$$

Step 1 (the representing vector). Let $(k,Z,m,A)$ be a Gaussian representation of $X$. By Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (iv), $Z\in\mathcal{L}^{2}(P;\mathbb{R}^{k})$ with $\mathbb{E}Z=0$ and $\operatorname{Cov}(Z)=I_{k}$. Let $U=AZ+m$, which is the random vector $m+AZ$. By Rules for Expectations, Variances and Covariances (xiii), $U\in\mathcal{L}^{2}(P;\mathbb{R}^{d})$, and with Rules for Expectations, Variances and Covariances (vi) and the rules $A0=0$ and $AI_{k}=A$ of Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv),

$$\mathbb{E}U=A\,\mathbb{E}Z+m=m,\qquad\operatorname{Cov}(U)=A\,I_{k}\,A^{\top}=AA^{\top}.$$

Step 2 ((ii)). By Gaussian Random Vectors and Jointly Gaussian Random Variables (i), $X=m+AZ=U$ almost surely, so Step 0 and Step 1 give $X\in\mathcal{L}^{2}(P;\mathbb{R}^{d})$, $\mathbb{E}X=m$ and $\operatorname{Cov}(X)=AA^{\top}$.

Step 3 ((iii)). Let $z:\Omega\to\mathbb{R}^{d}$ be the zero map. It is measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii)), and $|z|^{2}$ is the zero map, whose integral is $0$ (Basic Properties of the Integral of Nonnegative Measurable Maps (v)); so $z\in\mathcal{L}^{2}(P;\mathbb{R}^{d})$. Since $0_{d}\,z(\omega)+m=m$ for every $\omega$ (Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v)), the constant random vector $m$ is $0_{d}\,z+m$, and it lies in $\mathcal{L}^{2}(P;\mathbb{R}^{d})$ by Rules for Expectations, Variances and Covariances (xiii). Its coordinates are the constants $m_{i}$, with $\mathbb{E}\,m_{i}=m_{i}$ by Rules for Expectations, Variances and Covariances (ii), so its mean is $m$. Its covariance matrix has the entries $\operatorname{Cov}(m_{i},m_{j})=0$, as the covariance of a square-integrable random variable with a constant vanishes (Rules for Expectations, Variances and Covariances (x)); so it is $0_{d}$. If $X=m$ almost surely, Step 0 transfers these facts to $X$.

Step 4 ((i)). By Gaussian Random Vectors and Jointly Gaussian Random Variables (i), $X=m$ almost surely for some $m\in\mathbb{R}^{d}$, or $X$ has a Gaussian representation; so $X\in\mathcal{L}^{2}(P;\mathbb{R}^{d})$ by Step 3 or Step 2.