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

$d_0 |t|^{-\beta} \leq \varphi_X(t) \leq d_1 |t|^{-\beta} \quad \text{as } t\to\infty$

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 ^[1] if its characteristic function satisfies

$d_0 |t|^{\beta_0}\exp\big(-|t|^\beta/\gamma\big) \leq \varphi_X(t) \leq d_1 |t|^{\beta_1}\exp\big(-|t|^\beta/\gamma\big) \quad \text{as } t\to\infty$

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

^ ^a ^b 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.