An information complexity index for probability measures on ℝ with all moments

Abstract

We prove that, each probability meassure on [Formula: see text], with all moments, is canonically associated with (i) a ∗-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index [Formula: see text] consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec. 4.1 motivates this name). The class [Formula: see text] coincides with the [Formula: see text]-measures in the finite support case and includes the semi-circle laws in the infinite support case. In the infinite support case, the class [Formula: see text] includes the arcsine laws, and the class [Formula: see text] appeared in central limit theorems of quantum random walks in the sense of Konno. The classes [Formula: see text], with [Formula: see text], do not seem to be present in the literature. The class [Formula: see text] includes the Gaussian and Poisson measures and the associated ∗-Lie algebra is the Heisenberg algebra. The class [Formula: see text] includes the non-standard (i.e. neither Gaussian nor Poisson) Meixner distributions and the associated ∗-Lie algebra is a central extension of [Formula: see text]. Starting from [Formula: see text], the ∗-Lie algebra associated to the class [Formula: see text] is infinite dimensional and the corresponding classes include the higher powers of the standard Gaussian.