Subexponential distribution (light-tailed)

In probability theory, one definition of a subexponential distribution is as a probability distribution whose tails decay at an exponential rate, or faster: a real-valued distribution ${\cal {D}}$ is called subexponential if, for a random variable $X\sim {\cal {D}}$ ,

{\mathbb {P}}(|X|\geq x)=O(e^{-Kx})

, for large

x

and some constant

K>0

.

The subexponential norm, $\|\cdot \|_{\psi _{1}}$ , of a random variable is defined by

\|X\|_{\psi _{1}}:=\inf \ \{K>0\mid {\mathbb {E}}(e^{|X|/K})\leq 2\},

where the infimum is taken to be

+\infty

if no such

K

exists.

This is an example of a Orlicz norm. An equivalent condition for a distribution ${\cal {D}}$ to be subexponential is then that $\|X\|_{\psi _{1}}<\infty .$ ^[1]^: §2.7

Subexponentiality can also be expressed in the following equivalent ways:^[1]^: §2.7

${\mathbb {P}}(|X|\geq x)\leq 2e^{-Kx},$ for all $x\geq 0$ and some constant $K>0$ .
${\mathbb {E}}(|X|^{p})^{1/p}\leq Kp,$ for all $p\geq 1$ and some constant $K>0$ .
For some constant $K>0$ , ${\mathbb {E}}(e^{\lambda |X|})\leq e^{K\lambda }$ for all $0\leq \lambda \leq 1/K$ .
${\mathbb {E}}(X)$ exists and for some constant $K>0$ , ${\mathbb {E}}(e^{\lambda (X-{\mathbb {E}}(X))})\leq e^{K^{2}\lambda ^{2}}$ for all $-1/K\leq \lambda \leq 1/K$ .
${\sqrt {|X|}}$ is sub-Gaussian.