The notion, introduced by Jean-Pierre Kahane, is as follows:
A random variable with finite mean is sub-Gaussian if there is a positive number such that:
Such a constant is called a proxy variance, and we say that is -sub-Gaussian. If is sub-Gaussian, one is usually interested in the optimal proxy variance:
Note that the variance always gives a lower bound on the optimal proxy variance: . In particular, when , is said to be strictly sub-Gaussian.
The sub-Gaussian property is closely related to the tails of the distribution. Intuitively, being sub-Gaussian amounts to having tails lighter than a Gaussian. This is actually a characterization of the property. Let . Then:
That equivalence clearly implies exponential upper bounds for the tails of the distribution since a Gaussian satisfies
That can also be seen directly: for a -sub-Gaussian variable ,
The polynomial function is minimized on at , for which we obtain
In that sense, the sub-Gaussian property of any compactly supported random variable comes for free since in that case the tails are obviously lighter than those of a Gaussian. A simple general proxy variance is given by Hoeffding’s lemma. Let be supported on with . Then for any ,
so is -sub-Gaussian.
Back to the Beta where , this shows the Beta is -sub-Gaussian. The question of finding the optimal proxy variance is a more challenging issue. In addition to characterizing the optimal proxy variance of the Beta distribution in the note, we provide the simple upper bound . It matches with Hoeffding’s bound for the extremal case , , where the Beta random variable concentrates on the two-point set (and when Hoeffding’s bound is tight).