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 is called subexponential if, for a random variable ,

The subexponential norm, , of a random variable is defined by

This is an example of a Orlicz norm. An equivalent condition for a distribution to be subexponential is then that

Subexponentiality can also be expressed in the following equivalent ways:

1. for all and some constant .
2. for all and some constant .
3. For some constant , for all .
4. exists and for some constant , for all .
5. is sub-Gaussian.

References

• High-Dimensional Statistics: A Non-Asymptotic Viewpoint, Martin J. Wainwright, Cambridge University Press, 2019, .