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We propose the notion of  sub‐Weibull  distributions, which are characterized by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalizes the sub‐Gaussian and sub‐Exponential families to potentially heavier tailed distributions. Sub‐Weibull distributions are parameterized by a positive tail index   θ   and reduce to sub‐Gaussian distributions for   θ  =  1  /  2   and to sub‐Exponential distributions for   θ  =  1  . A characterization 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.

Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions