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 $$X\sim {\cal D} $$,
 * $${\Bbb P}(|X|\ge x)=O(e^{-K x}) $$, for large $$x$$ and some constant $$K>0$$.

The subexponential norm, $$\|\cdot\|_{\psi_1}$$, of a random variable is defined by
 * $$\|X\|_{\psi_1}:=\inf\ \{ K>0\mid {\Bbb E}(e^{|X|/K})\le 2\},$$ 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 $$\|X\|_{\psi_1}<\infty.$$

Subexponentiality can also be expressed in the following equivalent ways:


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