Sub-Gaussian variables and vectors, and sub-exponential variables

Use the real random variable convention of Random elements and real random variables and the expectation convention of Expectation of a nonnegative or integrable random variable. A real random variable $X$ is centered when it is integrable and $\mathbb{E}X=0$. A vector random element $Z=(Z_1,\dots ,Z_d)$ in ${\mathbb{R}}^d$, for an integer $d\ge 1$, is centered when each coordinate is integrable and $\mathbb{E}{Z}_j=0$. For $u,z\in {\mathbb{R}}^d$, write $⟨u,z⟩={\sum }_{j=1}^d{u}_j{z}_j$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_2=({\sum }_{j=1}^d{u}_j^2{)}^{1\mathrm{/}2}$, using the Euclidean convention in Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$.

For ${\sigma }^2\ge 0$, a centered real random variable $X$ is sub-Gaussian with proxy ${\sigma }^2$ if $\mathbb{E}{e}^{\lambda X}\le e^{{\lambda }^2{\sigma }^2\mathrm{/}2}\text{for every }\lambda \in \mathbb{R}\mathrm{.}$ A centered vector random element $Z\in {\mathbb{R}}^d$ is sub-Gaussian with proxy ${\sigma }^2$ if $\mathbb{E}{e}^{\lambda ⟨u,Z⟩}\le e^{{\sigma }^2{\lambda }^2\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\mathrm{/}2}\text{for every }u\in {\mathbb{R}}^d\text{ and }\lambda \in \mathbb{R}\mathrm{.}$

For ${\nu }^2\ge 0$ and $b>0$, a centered real random variable $X$ is sub-exponential with parameters $({\nu }^2,b)$ if $\mathbb{E}{e}^{\lambda X}\le e^{{\lambda }^2{\nu }^2\mathrm{/}2}\text{whenever }\mathrm{\mid }\lambda \mathrm{\mid }\le 1\mathrm{/}b\mathrm{.}$

The parameters ${\sigma }^2$ and ${\nu }^2$ are moment-generating-function upper-bound proxies; they are not asserted to equal the variance. The vector condition controls every one-dimensional projection and does not assert independence of coordinates.