Proof of The Density of a Nondegenerate Gaussian Law

Routine facts about real numbers and maps are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, Toolkit for CA1c, The Structure of ℝ as a Field and an Ordered Ring (ix)), and so is the agreement of the earlier and current versions of the Borel $\sigma$-algebra, measures and the integral of nonnegative functions (The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (v)). The idea is to realise $N(m,\Sigma)$ as the image of the standard normal law $\gamma_{k}$ under the affine map $z\mapsto Sz+m$, where $SS^{\top}=\Sigma$, and to pull the density $\varphi_{k}$ of $\gamma_{k}$ back by the linear change of variables.

A factor of $\Sigma$. Let $S=\Sigma^{1/2}$ be the positive semidefinite square root of $\Sigma$ (it exists because $\Sigma$ is positive semidefinite, Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (x)); it is symmetric with $SS=\Sigma$, so $SS^{\top}=\Sigma$. By The Determinant of a Positive Definite Matrix Is Positive (ii), applied to $A=S$, the matrix $S$ is invertible and $|\det S|=\sqrt{\det\Sigma}$. The rules for inverses of products and transposes (Rules for Matrix Inverses (iv), Rules for Matrix Inverses (v)) then give

$$(S^{-1})^{\top}S^{-1}=\bigl(SS^{\top}\bigr)^{-1}=\Sigma^{-1}.$$

$N(m,\Sigma)$ as an image of $\gamma_{k}$. On the probability space $(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}),\gamma_{k})$ (Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (i)) the identity map $Z$ is a standard normal vector (Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (ii)). Let $T:\mathbb{R}^{k}\to\mathbb{R}^{k}$ be the map $T(z)=Sz+m$; it is measurable (Linear Change of Variables for Lebesgue Measure on ℝⁿ (ii); this result, like its clause used in Pulling the density back, rests on dependent choice). The random vector $X=m+SZ$, which is the map $T$ itself, is Gaussian with Gaussian representation $(k,Z,m,S)$ (Gaussian Random Vectors and Jointly Gaussian Random Variables (i), Gaussian Random Vectors and Jointly Gaussian Random Variables (ii)), so its mean and covariance are

$$\mathbb{E}X=m,\qquad\operatorname{Cov}(X)=SS^{\top}=\Sigma$$

(Square-Integrability, Mean and Covariance Matrix of a Gaussian Random Vector (ii)). Being determined by its mean and covariance, the law of $X$ is $P_{X}=N(m,\Sigma)$ (A Random Vector Is Gaussian with Mean m and Covariance Σ Exactly When Its Law Is N(m,Σ) (i)). Since the law $P_{X}$ is the image measure of $\gamma_{k}$ under $X=T$ (The Law of a Random Element (i)),

$$N(m,\Sigma)(B)=\gamma_{k}\bigl(T^{-1}(B)\bigr)\qquad(B\in\mathcal{B}(\mathbb{R}^{k})).$$

Pulling the density back. Fix $B\in\mathcal{B}(\mathbb{R}^{k})$. Since $\gamma_{k}$ is the measure with density $\varphi_{k}$ with respect to $\lambda_{k}$ (Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (ii), Measure with a Density (i)) and $\mathbf{1}_{T^{-1}(B)}=\mathbf{1}_{B}\circ T$, writing the integral over $T^{-1}(B)$ with an indicator (Integrals over Measurable Subsets (i)) gives

$$\gamma_{k}\bigl(T^{-1}(B)\bigr)=\int_{\mathbb{R}^{k}}(\mathbf{1}_{B}\circ T)\,\varphi_{k}\,d\lambda_{k}.$$

To apply the change of variables we write the integrand as a function composed with $T$. The inverse of $T$ is the map $U(y)=S^{-1}(y-m)=S^{-1}y-S^{-1}m$ (Rules for Matrix Inverses (vi), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v)). As $S^{-1}$ is invertible (Rules for Matrix Inverses (iii)) and $\varphi_{k}\in\mathcal{M}^{+}(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$ (Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (ii)), the change of variables applied to the affine map $U$ (Linear Change of Variables for Lebesgue Measure on ℝⁿ (iii), with $A=S^{-1}$ and $b=-S^{-1}m$) shows that the function $h=\varphi_{k}\circ U$, $h(y)=\varphi_{k}(S^{-1}(y-m))$, lies in $\mathcal{M}^{+}(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$, and hence so does

$$g=h\,\mathbf{1}_{B},\qquad g(y)=\mathbf{1}_{B}(y)\,\varphi_{k}\bigl(S^{-1}(y-m)\bigr)$$

(Integrals over Measurable Subsets (i)); moreover $g\circ T=(\mathbf{1}_{B}\circ T)\,\varphi_{k}$, because $U\circ T$ is the identity. Put

$$c=\bigl(\sqrt{\det\Sigma}\bigr)^{-1}=|\det S|^{-1},$$

using the value of $|\det S|$ from A factor of $\Sigma$. This is a positive real number, because $\det\Sigma>0$ (The Determinant of a Positive Definite Matrix Is Positive (i)), so $ch\in\mathcal{M}^{+}(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$ by homogeneity (Basic Properties of the Integral of Nonnegative Measurable Maps (iv)). Now

$$\begin{aligned} N(m,\Sigma)(B)&=\int_{\mathbb{R}^{k}}g\circ T\,d\lambda_{k}=|\det S|^{-1}\int_{\mathbb{R}^{k}}g\,d\lambda_{k}=c\int_{\mathbb{R}^{k}}h\mathbf{1}_{B}\,d\lambda_{k}\\ &=\int_{\mathbb{R}^{k}}(ch)\mathbf{1}_{B}\,d\lambda_{k}=\int_{B}ch\,d\lambda_{k}, \end{aligned}$$

by the linear change of variables (Linear Change of Variables for Lebesgue Measure on ℝⁿ (iii)), the definitions of $g$ and $c$, homogeneity (Basic Properties of the Integral of Nonnegative Measurable Maps (iv)), as $(ch)\mathbf{1}_{B}=c\,(h\mathbf{1}_{B})$, and the indicator form of the integral over $B$ (Integrals over Measurable Subsets (i)).

Identifying the integrand. By The Standard Normal Densities in Closed Form (ii), for $z\in\mathbb{R}^{k}$,

$$\varphi_{k}(z)=\frac{1}{(\sqrt{2\pi})^{k}}\exp\bigl(-\lVert z\rVert^{2}/2\bigr).$$

For $z=S^{-1}(y-m)$ we have $\lVert z\rVert^{2}=z\cdot z$ (Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i)), and moving $S^{-1}$ across the dot product (Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (viii), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v)) and then using A factor of $\Sigma$ give

$$\bigl\lVert S^{-1}(y-m)\bigr\rVert^{2}=(y-m)\cdot(S^{-1})^{\top}S^{-1}(y-m)=(y-m)\cdot\Sigma^{-1}(y-m).$$

The two factors in the denominator of $f$ are nonzero: $(\sqrt{2\pi})^{k}\neq0$, as $2\pi>0$ (Quarter-Turn Identities and Periodicity of Sine and Cosine (i)), and $\sqrt{\det\Sigma}\neq0$, as $\det\Sigma>0$. So the reciprocal of their product is the product of their reciprocals (Rules of Arithmetic and Order in an Ordered Field (iii)), and for every $y\in\mathbb{R}^{k}$

$$f(y)=\frac{1}{\sqrt{\det\Sigma}}\cdot\frac{1}{(\sqrt{2\pi})^{k}}\exp\Bigl(-\tfrac{1}{2}\bigl\lVert S^{-1}(y-m)\bigr\rVert^{2}\Bigr)=c\,\varphi_{k}\bigl(S^{-1}(y-m)\bigr)=c\,h(y),$$

with $h=\varphi_{k}\circ U$ from Pulling the density back. Hence $f=ch$, so $f\in\mathcal{M}^{+}(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$ and $N(m,\Sigma)(B)=\int_{B}f\,d\lambda_{k}$ for every $B\in\mathcal{B}(\mathbb{R}^{k})$, which is (i).