Definition:Sub-Exponential Distribution

Definition
The distribution of a random variable $X$ with expectation $\mu = \expect X$ is called sub-exponential if there exists $\nu, \alpha \in \R_{\ge 0}$ such that:


 * $\expect {e^{\lambda \paren {X - \mu} } } \le e^{\nu^2 \lambda^2 / 2}$

for all $\size \lambda < \dfrac 1 \alpha$.

Basic Properties
The sub-Gaussian distribution results as a special case ($\nu = \sigma, \alpha = 0$).