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

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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≤ q ≤ 1, almost surely.