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

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

The subexponential norm, $\Vert \cdot {\Vert }_{{\psi }_1}$, of a random variable is defined by

$$\begin{gathered} \Vert X{\Vert }_{{\psi }_1}:=\inf \\ \{K>0\mid \mathbb{𝔼}(e^{|X|/K})\le 2\}, \end{gathered}$$ where the infimum is taken to be $+\mathrm{\infty }$ if no such $K$ exists.

This is an example of a Orlicz norm. An equivalent condition for a distribution ${D}$ to be subexponential is then that $\Vert X{\Vert }_{{\psi }_1}<\mathrm{\infty }.$ Subexponentiality can also be expressed in the following equivalent ways:

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

References

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