Definition:Sub-Exponential Distribution

Definition

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

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

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

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Basic Properties

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