Modified Stieltjes Transform and Generalized Convolutions of Probability Distributions

Abstract

The classical Stieltjes transform is modified in such a way as to generalize both Stieltjes and Fourier transforms. This transform allows the introduction of new classes of commutative and non-commutative generalized convolutions.

A particular case of such a convolution for degenerate distributions appears to be the Wigner semicircle distribution.

Keywords

• Stieltjes transform
• characteristic function
• generalized convolution
• beta distribution