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Smoothness (probability theory)

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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 | − β ≤ | φ X ( t ) | ≤ d 1 | t | − β as  t → ∞ {\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 if its characteristic function satisfies

d 0 | t | β 0 exp ⁡ ( − | t | β / γ ) ≤ | φ X ( t ) | ≤ d 1 | t | β 1 exp ⁡ ( − | t | β / γ ) as  t → ∞ {\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. ^ 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.


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