As a follow-up on my previous post on the sub-Gaussian property for the Beta distribution , I’ll give here a visual illustration of the proof.
A random variable with finite mean is sub-Gaussian if there is a positive number such that:
We focus on X being a Beta random variable. Its moment generating function is known as the Kummer function, or confluent hypergeometric function . So X is -sub-Gaussian as soon as the difference function
remains positive on . This difference function is plotted on the right panel above for parameters . In the plot, is varying from green for the variance (which is a lower bound to the optimal proxy variance) to blue for the value , a simple upper bound given by Elder (2016), . The idea of the proof is simple: the optimal proxy-variance corresponds to the value of for which admits a double zero, as illustrated with the red curve (black dot). The left panel shows the curves with varying, interpolating from green for to blue for , with only one curve qualifying as the optimal proxy variance in red.
 Marchal and Arbel (2017), On the sub-Gaussianity of the Beta and Dirichlet distributions. Electronic Communications in Probability, 22:1–14, 2017. Code on GitHub.
 Elder (2016), Bayesian Adaptive Data Analysis Guarantees from Subgaussianity, https://arxiv.org/abs/1611.00065