 Multivariate Gaussian distribution

A multivariate Gaussian distribution is the law of a Random vector $X\in {\mathbb{R}}^d$ such that every linear combination $a^{T}X$ has a Normal distribution, allowing a constant as a zero-variance Normal distribution. Its Mean vector $\mu$ and positive-semidefinite covariance matrix $\Sigma$ determine its Characteristic function:If $\Sigma$ is positive definite, the law has the normalized Probability density functionA singular covariance matrix is also permitted, but this density formula is not: the law is concentrated on the affine subspace $\mu +\mathrm{range}\Sigma$. For example, $(Z,Z)$ with $Z$ standard normal has covariance $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ and lives on the diagonal, so it has no two-dimensional Lebesgue density.

A construction valid in both cases is $X=\mu +AZ$, where $Z$ has independent standard normal coordinates and $A{A}^T=\Sigma$. Its Characteristic function follows by multiplying the one-dimensional characteristic functions. For $\mathrm{rank}\Sigma =r$, one can use a matrix $A$ with $r$ columns.

Normal marginals alone do not imply a multivariate Gaussian law. Let $Z$ be standard normal and let $S$ be an independent Rademacher random variable. Both $Z$ and $SZ$ are standard normal, but their sum is zero with probability $1/2$ and otherwise equals $2Z$, so that nonconstant sum is not normal. Joint Gaussianity is the stronger condition needed for Gaussian moment and linear-transformation rules.