Sub-Gaussian distribution

In probability theory, a sub-Gaussian distribution, the distribution of a sub-Gaussian random variable, is a probability distribution with strong tail decay. More specifically, the tails of a sub-Gaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gives sub-Gaussian distributions their name.

Formally, the probability distribution of a random variable ${\displaystyle X}$ is called sub-Gaussian if there is a positive constant C such that for every ${\displaystyle t\geq 0}$,

${\textstyle \operatorname {P} (|X|\geq t)\leq 2\exp {(-t^{2}/C^{2})}}$.

Alternatively, a random variable is considered sub-Gaussian if its distribution function is upper bounded (up to a constant) by the distribution function of a Gaussian. Specifically, we say that ${\displaystyle X}$ is sub-Gaussian if for all ${\displaystyle s\geq 0}$ we have that:

${\displaystyle P(|X|\geq s)\leq cP(|Z|\geq s),}$

where ${\displaystyle c\geq 0}$ is constant and ${\displaystyle Z}$ is a mean zero Gaussian random variable.^{: Theorem 2.6}