Sub-Gaussian distribution

A centered random variable is sub-Gaussian with parameter $\sigma$ when $\mathbb{E}{e}^{tX}\le e^{{\sigma }^2{t}^2/2}$ for every real $t$.

A centered random vector is sub-Gaussian with variance proxy ${\tau }^2$ when the displayed inequality holds for every deterministic vector $a$. Equivalently every scalar inner product is a sub-Gaussian random variable with proxy ${\tau }^2\Vert a{\Vert }_2^2$. Independence of coordinates is not required. For vectors $a_j$ with Euclidean norms at most one, the union bound gives ${\max }_j|⟨a_j,g⟩|\le \tau \sqrt{2\log (2m/\delta )}$ with probability at least $1-\delta$. This assumption alone is weaker than concentration for all Lipschitz functions.