Subexponential distribution (light-tailed)

Template:Short description Template:Refimprove In probability theory, in the context of light-tailed distributions, 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 $X\sim 𝒟$,

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

Note that this is almost the opposite of the more established meaning of subexponential in the context of Heavy-tailed distributions, where "sub" means that the rate of decay is slower than exponential, rather than that the tail is lighter than exponential.

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

$\Vert X{\Vert }_{{\psi }_1}:=inf\{K>0\mid 𝔼(e^{|X|/K})\le 2\},$ 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 $\Vert X{\Vert }_{{\psi }_1}<\mathrm{\infty }.$Template:R

Subexponentiality can also be expressed in the following equivalent ways:Template:R

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

Further reading

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