Smoothness (probability theory) explained

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

d _{0} |t| ^{-\beta} \leq |\varphi _{X(t)|} \leq d _{1} |t| ^{-\beta} as t\to inf ty 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

d _{0} |t| ^{\beta _{0}} \exp ( - |t| ^{\beta/\gamma)} \leq |\varphi _{X(t)|} \leq d _{1} |t| ^{\beta _{1}} \exp ( - |t| ^{\beta/\gamma)} as t\to inf ty 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

• Book: Lighthill , M. J. . 1962 . Introduction to Fourier analysis and generalized functions . London: Cambridge University Press .

Notes and References

1. Fan. Jianqing. 1991. On the optimal rates of convergence for nonparametric deconvolution problems. The Annals of Statistics. 19. 3. 1257–1272. 2241949. 10.1214/aos/1176348248. free.