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 ¹ if its characteristic function satisfies

${\displaystyle d_{0}|t|^{-\beta }\leq |\varphi _{X}(t)|\leq d_{1}|t|^{-\beta }\quad {\text{as }}t\to \infty }$

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 ¹ if its characteristic function satisfies

${\displaystyle d_{0}|t|^{\beta _{0}}\exp {\big (}-|t|^{\beta }/\gamma {\big )}\leq |\varphi _{X}(t)|\leq d_{1}|t|^{\beta _{1}}\exp {\big (}-|t|^{\beta }/\gamma {\big )}\quad {\text{as }}t\to \infty }$

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.