Central limit theorems for Coxeter systems and Artin systems of extra large type

Given a Coxeter system of large type we prove a non–commutative central limit theorem: After normalisation with the square root of n the characteristic function of the set of the first n generators tends in distribution to Wigners semi–circle law. If one chooses the group presentation at random, now admitting a certain amount of commutation among the generators, the distributional limit will be the q–deformed semi–circle law, $1\le q\le 1$, almost surely.