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Sketches on dilation probability distributions

Wesołowski J.

Demonstratio Mathematica

2001

Vol. 34, nr 2

385-402

The dilation property allows to define an intriguing family of statistical distributions parameterized by the coefficients of respective dilation equation and the dilation scale. The family includes, except some commonly used probability laws, also a wide range of naturally arising singular distributions, which usually are difficult for statistical analysis. But here due to dilation scheme some progress in developing statistical tools can be expected. The paper describes basic properties of dilation distributions, including an extension of the Kershner-Wintner theorem on infinite Bernoulli convolutions, and indicates possible directions for future studies, including preliminary observations on -tatistical inference.