## ⓘ 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

d 0 | t | − β ≤ φ X t ≤ d 1 | t | − β as t → ∞ {\displaystyle d_{0}|t|^{-\beta }\leq \varphi _{X}t\leq d_{1}|t|^{-\beta }\quad {\text{as }}t\to \infty }for some positive constants d 0, d 1, β. The examples of such distributions are gamma, exponential, uniform, etc.

The distribution is called supersmooth of order β if its characteristic function satisfies

d 0 | t | β 0 exp − | t | β / γ ≤ φ X t ≤ d 1 | t | β 1 exp − | t | β / γ as t → ∞ {\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 d 0, d 1, β, γ and constants β 0, β 1. Such supersmooth distributions have derivatives of all orders. Examples: normal, Cauchy, mixture normal.

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