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 {{cite journal|last=Fan|first=Jianqing|year=1991|title=On the optimal rates of convergence for nonparametric deconvolution problems|journal=The Annals of Statistics|volume=19|issue=3|pages=1257–1272|jstor=2241949|doi=10.1214/aos/1176348248|doi-access=free}} if its characteristic function satisfies

: $d\_0\; |t|^\{-\backslash beta\}\; \backslash leq\; |\backslash varphi\_X(t)|\; \backslash leq\; d\_1\; |t|^\{-\backslash beta\}\; \backslash quad\; \backslash text\{as\; \}\; t\backslash to\backslash infty$

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 if its characteristic function satisfies

: $d\_0\; |t|^\{\backslash beta\_0\}\backslash exp\backslash big(-|t|^\backslash beta/\backslash gamma\backslash big)\; \backslash leq\; |\backslash varphi\_X(t)|\; \backslash leq\; d\_1\; |t|^\{\backslash beta\_1\}\backslash exp\backslash big(-|t|^\backslash beta/\backslash gamma\backslash big)\; \backslash quad\; \backslash text\{as\; \}\; t\backslash to\backslash infty$

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.