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 [math]\displaystyle{ \cal D }[/math] is called subexponential if, for a random variable [math]\displaystyle{ X\sim {\cal D} }[/math],

[math]\displaystyle{ {\Bbb P}(|X|\ge x)=O(e^{-K x}) }[/math], for large [math]\displaystyle{ x }[/math] and some constant [math]\displaystyle{ K\gt 0 }[/math].

The subexponential norm, [math]\displaystyle{ \|\cdot\|_{\psi_1} }[/math], of a random variable is defined by

[math]\displaystyle{ \|X\|_{\psi_1}:=\inf\ \{ K\gt 0\mid {\Bbb E}(e^{|X|/K})\le 2\}, }[/math] where the infimum is taken to be [math]\displaystyle{ +\infty }[/math] if no such [math]\displaystyle{ K }[/math] exists.

This is an example of a Orlicz norm. An equivalent condition for a distribution [math]\displaystyle{ \cal D }[/math] to be subexponential is then that [math]\displaystyle{ \|X\|_{\psi_1}\lt \infty. }[/math]

Subexponentiality can also be expressed in the following equivalent ways:

1. [math]\displaystyle{ {\Bbb P}(|X|\ge x)\le 2 e^{-K x}, }[/math] for all [math]\displaystyle{ x\ge 0 }[/math] and some constant [math]\displaystyle{ K\gt 0 }[/math].
2. [math]\displaystyle{ {\Bbb E}(|X|^p)^{1/p}\le K p, }[/math] for all [math]\displaystyle{ p\ge 1 }[/math] and some constant [math]\displaystyle{ K\gt 0 }[/math].
3. For some constant [math]\displaystyle{ K\gt 0 }[/math], [math]\displaystyle{ {\Bbb E}(e^{\lambda |X|}) \le e^{K\lambda} }[/math] for all [math]\displaystyle{ 0\le \lambda \le 1/K }[/math].
4. [math]\displaystyle{ {\Bbb E}(X) }[/math] exists and for some constant [math]\displaystyle{ K\gt 0 }[/math], [math]\displaystyle{ {\Bbb E}(e^{\lambda (X-{\Bbb E}(X))})\le e^{K^2 \lambda^2} }[/math] for all [math]\displaystyle{ -1/K\le \lambda\le 1/K }[/math].
5. [math]\displaystyle{ \sqrt{|X|} }[/math] is sub-Gaussian.

References

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