Subexponential distribution (light-tailed) explained

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 {\cal D} , {\Bbb P}(|X|\ge x) = O(e ^{-K} ) , for large x and some constant K> 0 .The subexponential norm, \| ⋅ \| _{\psi _{1}} , of a random variable is defined by\|X\| _{\psi _{1}} : =inf \{ K> 0 \mid {\Bbb E}(e ^{|X|/K} )\le 2 \}, where the infimum is taken to be +inf ty 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\| _{\psi _{1}} < inf ty.

Subexponentiality can also be expressed in the following equivalent ways:

{\Bbb P}(|X|\ge x)\le 2 e ^{-K} , for all x\ge 0 and some constant K> 0 .{\Bbb E}(|X| ^{p)} ^{1/p} \le K p, for all p\ge 1 and some constant K> 0 .

1. For some constant

K> 0 , {\Bbb E}(e ^{λ} ) \le e ^{Kλ} for all 0 \le λ \le 1 /K .{\Bbb E}(X) exists and for some constant K> 0 , {\Bbb E}(e ^{λ} (X))})\le e ^{K ^{2} λ ^{2}} for all -1 /K\le λ \le 1 /K .\sqrt{|X|} is sub-Gaussian.

References

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