The characterization of a class of probability measures by multiplicative renormalization

Abstract

We use the multiplicative renormalization method to characterize a class of probability measures on R determined by five parameters. This class of probability measures contains the arcsine and the Wigner semicircle distributions (the vacuum distributions of the field operators of interacting Fock spaces related to the Anderson model), as well as new nonsymmetric distributions. The corresponding orthogonal polynomials and Jacobi-Szegö parameters are derived from the orthogonal-polynomial generating functions. These orthogonal polynomials can be expressed in terms of the Chebyshev polynomials of the second kind.

On the non-classical infinite divisibility of power semicircle distributions

“…The case θ = 0 is the semicircle distribution on (−σ, σ) and the right-boundary case θ = ∞ given by Poincaré's theorem is the classical Gaussian distribution. Other important families of compactly supported distributions which are useful in non-commutative probability and that include the arcsine and semicircle distributions are considered in Kubo, Kuo and Namli [29], [30] and references therein.…”

Applicability of multiplicative renormalization method for a certain function

“…For convenience, we use the following definition from [8]. The probability measures such as Gaussian, Poisson, gamma, uniform, arcsine, semi-circle, beta, and Pascal are MRM-applicable for functions of the form…”

“…On the other hand, we have recently solved this problem for the case h(x) = (1−x) −1 in [8]. The class of MRM-applicable measures for h(x) = (1−x)…”

Classes of MRM-Applicable Measures for the Power Function of General Order

“…[9][10][11][12] Since the first two cases are extremal and have been discussed in previous papers, we will exclude them in this paper. On the contrary, the case κ = 2 can be included in the general cases.…”