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 ${\displaystyle {\cal {D}}}$ is called subexponential if, for a random variable ${\displaystyle X\sim {\cal {D}}}$,

${\displaystyle {\mathbb {P}}(|X|\geq x)=O(e^{-Kx})}$, for large ${\displaystyle x}$ and some constant ${\displaystyle K>0}$.

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

${\displaystyle \|X\|_{\psi _{1}}:=\inf \ \{K>0\mid {\mathbb {E}}(e^{|X|/K})\leq 2\},}$ where the infimum is taken to be ${\displaystyle +\infty }$ if no such ${\displaystyle K}$ exists.

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

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

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

References

• High-Dimensional Statistics: A Non-Asymptotic Viewpoint, Martin J. Wainwright, Cambridge University Press, 2019, ISBN 9781108498029.