Subexponential distribution (light-tailed)
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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 ,
- , for large and some constant .
The subexponential norm, , of a random variable is defined by
- where the infimum is taken to be if no such exists.
This is an example of a Orlicz norm. An equivalent condition for a distribution to be subexponential is then that [1]: §2.7
Subexponentiality can also be expressed in the following equivalent ways:[1]: §2.7
- for all and some constant .
- for all and some constant .
- For some constant , for all .
- exists and for some constant , for all .
- is sub-Gaussian.
References
- ^ a b High-Dimensional Probability: An Introduction with Applications in Data Science, Roman Vershynin, University of California, Irvine, June 9, 2020
- High-Dimensional Statistics: A Non-Asymptotic Viewpoint, Martin J. Wainwright, Cambridge University Press, 2019, ISBN 9781108498029.