Limit theorems on quantum groups

Abstract

Limit theorems are a fundamental part of probability theory. Such theorems have been proved over the last decades for Lie groups [GKR77] and commutative hy-pergroups [BH95] but only few limit theorems exist in the quantum group case. In this chapter we present different aspects of that theory. In Section 10.1, the general results for limit theorems on bialgebras due to Schürmann [Sch93] is presented. Then a randomised q-central (or q-commutative) limit theorem on a family of bialgebras with one complex parameter is shown in Section 10.2. This result is due to U. Franz [Pra98]. Section 10.3 is devoted to Woronowicz results [Wor87] on convergence of convolution products of probability measures to the Haar functional on compact quantum groups. A q-central limit theorem for U _q(su(2)) has been proved by R. Lenczewski [Len93, Len94], this result is stated in Section 10.4 as well as a weak law of large numbers which derives easily from the central limit theorem. Then we recall D. Neuenschwander and R. Schott’s (cf. [NS97]) results on domains of attraction for q-transformed random variables in Section 10.5.