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 ,

, 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 

  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

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  1. ^ 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.