In probability theory, in the context of light-tailed distributions, 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 . Note that this is almost the opposite of the more established meaning of subexponential in the context of Heavy-tailed distributions, where "sub" means that the rate of decay is slower than exponential, rather than that the tail is lighter than exponential.
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
Subexponentiality can also be expressed in the following equivalent ways:
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 High-Dimensional Probability: An Introduction with Applications in Data Science, Roman Vershynin, University of California, Irvine, June 9, 2020
Further reading High-Dimensional Statistics: A Non-Asymptotic Viewpoint, Martin J. Wainwright, Cambridge University Press, 2019, .