# Sub Gaussian

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This post is based on Wainwright (2019).

A random variable $X$ with mean $\mu=\E X$ is sub-gaussian if there is a positive number $\sigma$ such that
$$
\E [e^{\lambda(X-\mu)}]\le e^{\lambda^2\sigma^2/2}\;\forall \lambda\in \IR\,.
$$

Several equivalent characterizations of sub-Gaussian variables.

Given any zero-mean random variable $X$, the following properties are equivalent:

- There is a constant $\sigma$ such that $\E e^{\lambda X}\le e^{\lambda^2\sigma^2/2}$ for all $\lambda \in \IR$.
- There is a constant $c$ and Gaussian random variable $Z\sim N(0,\sigma^2)$ such that $P(\vert X\vert\ge s)\le cP(\vert Z\vert \ge s)$ for all $s\ge 0$
- There exists a number $\theta$ such that $\E X^{2k}\le\frac{(2k)!}{2^kk!}\theta^{2k}$ for all $k=1,2,\ldots$
- We have $\E e^{\frac{\lambda X^2}{2\sigma^2}}\le \frac{1}{\sqrt{1-\lambda}}$ for all $\lambda \in [0, 1)$.
- $\psi_2$-condition: $\exists a>0, \E e^{aX^2}\le 2$. Refer to Subgaussian random variables: An expository note