Introduction to High-Dimensional Statistics, 2nd Edition

Appendix A

Gaussian Distribution

A.1 Gaussian Random Vectors

A random vector Y ∈ ℝ^d is distributed according to the N (m, Σ) Gaussian distribution, with m ∈ ℝ^d and $\Sigma \in S_d^{+}$ (the set of all d × d symmetric positive semi-definite matrix), when

$\mathbb{E}\left[e^{i\left(\lambda ,Y\right)}\right]=\exp \left(i\langle \lambda ,m\rangle -\frac{1}{2}{\lambda }^T\Sigma \lambda \right),forall\lambda \in {ℝ}^d.$ (A.1)

When matrix Σ is non-singular (i.e., positive definite), the N (m, Σ) Gaussian distribution has a density with respect to the Lebesgue measure on ℝ^d given by

$\frac{1}{{\left(2\pi \right)}^{d/2}\det {\left(\Sigma \right)}^{1/2}}\exp \left(-\frac{1}{2}{\left(y-m\right)}^T{\Sigma }^{-1}\left(y-m\right)\right).$

Affine transformations of Gaussian distribution are still Gaussian.

Lemma A.1 Affine transformation

Let Y ∈ ℝ^d be a random vector with N (m, Σ) Gaussian ...