  We investigate the sub-Gaussian property for almost surely bounded random
variables. If sub-Gaussianity per se is de facto ensured by the bounded support
of said random variables, then exciting research avenues remain open. Among
these questions is how to characterize the optimal sub-Gaussian proxy variance?
Another question is how to characterize strict sub-Gaussianity, defined by a
proxy variance equal to the (standard) variance? We address the questions in
proposing conditions based on the study of functions variations. A particular
focus is given to the relationship between strict sub-Gaussianity and symmetry
of the distribution. In particular, we demonstrate that symmetry is neither
sufficient nor necessary for strict sub-Gaussianity. In contrast, simple
necessary conditions on the one hand, and simple sufficient conditions on the
other hand, for strict sub-Gaussianity are provided. These results are
illustrated via various applications to a number of bounded random variables,
including Bernoulli, beta, binomial, uniform, Kumaraswamy, and triangular
distributions.
