Limit theorems for the hierarchy of freeness

Uwe Franz (1) , Romuald Lenczewski

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

The central limit theorem, the invariance principle and the Poisson limit theorem for the hierarchy of freeness are studied. We show that for given natural m the limit laws can be expressed in terms of non-crossing partitions of depth smaller or equal to m. For the algebra of polynomials in one variable we solve the associated moment problems and find explicitly the discrete limit measures.

Domaines: Algèbres quantiques [math.QA]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Uwe Franz, Romuald Lenczewski. Limit theorems for the hierarchy of freeness. 2001. ⟨hal-00470217⟩