A one-parameter family of transforms, linearizing convolution laws for probability distributions

Abstract

We study a family of transforms, depending on a parameterq∈[0,1], which interpolate (in an algebraic framework) between a relative (namely: -iz(log ℱ(·)) ^′(-iz)) of the logarithm of the Fourier transform for probability distributions, and its free analogue constructed by D. Voiculescu ([16, 17]). The classical case corresponds toq=1, and the free one toq=0.

We describe these interpolated transforms: (a) in terms of partitions of finite sets, and their crossings; (b) in terms of weighted shifts; (c) by a matrix equation related to the method of Stieltjes for expanding continuedJ-fractions as power series. The main result of the paper is that all these descriptions, which extend basic approaches used forq=0 and/orq=1, remain equivalent for arbitraryq∈[0, 1].

We discuss a couple of basic properties of the convolution laws (for probability distributions) which are linearized by the considered family of transforms (these convolution laws interpolate between the usual convolution — atq=1, and the free convolution introduced by Voiculescu — atq=0). In particular, we note that description (c) mentioned in the preceding paragraph gives an insight of why the central limit law for the interpolated convolution has to do with theq-continuous Hermite orthogonal polynomials.