Supersmooth

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
: $d_0 , t, ^ \leq , \varphi_X(t), \leq d_1 , t, ^ \quad \text t\to\infty$
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
: $d_0 , t, ^\exp\big(-, t, ^\beta/\gamma\big) \leq , \varphi_X(t), \leq d_1 , t, ^\exp\big(-, t, ^\beta/\gamma\big) \quad \text t\to\infty$
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.

References

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Theory of probability distributions

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