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 $X\sim$,
:$(, X, \ge x)=O(e^)$, for large $x$ and some constant $K>0$.
The subexponential norm, $\, \cdot\, _$, of a random variable is defined by
:$\, X\, _:=\inf\ \,$ 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 $\, X\, _<\infty.$

Subexponentiality can also be expressed in the following equivalent ways:

# $(, X, \ge x)\le 2 e^,$ for all $x\ge 0$ and some constant $K>0$.
# $(, X, ^p)^\le K p,$ for all $p\ge 1$ and some constant $K>0$.
# For some constant $K>0$, $(e^) \le e^$ for all $0\le \lambda \le 1/K$.
# $(X)$ exists and for some constant $K>0$, $(e^)\le e^$ for all $-1/K\le \lambda\le 1/K$.
# $\sqrt$ is sub-Gaussian.

References

* ''High-Dimensional Statistics: A Non-Asymptotic Viewpoint'', Martin J. Wainwright, Cambridge University Press, 2019, .
Continuous distributions

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