Definition:Sub-Exponential Distribution

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


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

for all $|\lambda| < \frac{1}{\alpha}$.

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