Smoothness (probability theory)
In probability theory and statistics, smoothness of a density function is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s characteristic function.
Formally, we call the distribution of a random variable X ordinary smooth of order β if its characteristic function satisfies
for some positive constants d0, d1, β. The examples of such distributions are gamma, exponential, uniform, etc.
The distribution is called supersmooth of order β if its characteristic function satisfies
for some positive constants d0, d1, β, γ and constants β0, β1. Such supersmooth distributions have derivatives of all orders. Examples: normal, Cauchy, mixture normal.
