# 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.

## References

1. ^ a b Fan, Jianqing (1991). "On the optimal rates of convergence for nonparametric deconvolution problems". The Annals of Statistics. 19 (3): 1257–1272. doi:10.1214/aos/1176348248. JSTOR 2241949.
• Lighthill, M. J. (1962). Introduction to Fourier analysis and generalized functions. London: Cambridge University Press.