On probabilistic aspects of Chebyshev polynomials

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.

16 pages

References in corpus (3)

• Expansions of one density via polynomials orthogonal with respect to the other
• On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters
• On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures