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

${\displaystyle d_0|t{|}^{-\beta }\le |{\phi }_X(t)|\le d_1|t{|}^{-\beta }\text{as }t\to \mathrm{\infty }}$

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

${\displaystyle d_0|t{|}^{{\beta }_0}\exp (-|t{|}^{\beta }/\gamma )\le |{\phi }_X(t)|\le d_1|t{|}^{{\beta }_1}\exp (-|t{|}^{\beta }/\gamma )\text{as }t\to \mathrm{\infty }}$

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

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

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