On probabilistic aspects of Chebyshev polynomials

TL;DR

This paper introduces new multidimensional densities using Chebyshev polynomials, enabling effective calculation of moments and orthogonal polynomials, with potential applications in probabilistic modeling.

Contribution

It develops a novel method to construct multidimensional distributions based on Chebyshev polynomials and their properties, facilitating calculations and analysis.

Findings

Derived new multidimensional densities with compact support

Calculated all moments of the introduced distributions

Identified orthogonal polynomial families related to these distributions

Abstract

} The main goal of this note is to provide new, mostly multidimensional densities, compactly supported and list many of its properties that enable effective calculations. The idea of obtaining such densities is firstly to build some one-dimensional densities depending on many parameters and then treat the constructed in this way distributions as conditional ones. Then of course by imposing certain distribution on the parameters and multiplying the two distributions we arrive at new multivariate distribution. To enable effective calculations, we utilize nice, simple and widely known properties of Chebyshev polynomials. Thus, in particular, the one-dimensional distribution mentioned above will have a form of arcsine distribution multiplied by some rational function. The fact that we use Chebyshev polynomials allows us to calculate all moments of this one-dimensional distribution as well as to find a family of polynomials that are orthogonal with respect to this distribution.