A random integral calculus on generalized s-selfdecomposable probability measures

TL;DR

This paper explores the properties of generalized s-selfdecomposable probability measures, showing that compositions of certain integral mappings result in new mappings with a modified inner time, and provides related distribution formulas.

Contribution

It demonstrates that compositions of random integral mappings on generalized s-selfdecomposable measures produce new mappings with a modified inner time, extending the understanding of their structure.

Findings

Composition of mappings results in a new integral mapping with a different inner time.

Provides formulas for distributions of products of powers of independent uniform variables.

Uses Lagrange interpolation in the proof.

Abstract

It is known that the class $\mathcal{U}_{\beta}$, of generalized s-selfdecom-posable probability distributions, can be viewed as an image via random integral mapping $\mathcal{J}^{\beta}$ of the class $ID$ of all infinitely divisible measures. We prove that a composition of the mappings $\mathcal{J}^{\beta_1}, \mathcal{J}^{\beta_2}, ..., \mathcal{J}^{\beta_n}$ is again random integral mapping but with a new inner time. In a proof some form of Lagrange interpolation formula is needed. Moreover, some elementary formulas concerning the distributions of products of powers of independent uniformly distributed random variables as established as well.