  We propose the notion of sub-Weibull distributions, which are characterised
by tails lighter than (or equally light as) the right tail of a Weibull
distribution. This novel class generalises the sub-Gaussian and sub-Exponential
families to potentially heavier-tailed distributions. Sub-Weibull distributions
are parameterized by a positive tail index $\theta$ and reduce to sub-Gaussian
distributions for $\theta=1/2$ and to sub-Exponential distributions for
$\theta=1$. A characterisation of the sub-Weibull property based on moments and
on the moment generating function is provided and properties of the class are
studied. An estimation procedure for the tail parameter is proposed and is
applied to an example stemming from Bayesian deep learning.
