Central limit theorems for permutation statistics

The number of inversions or descents of a random permutation in a large symmetric group is asymptotically normally distributed. We discuss extensions of this principle to arbitrary families of finite Coxeter groups of increasing rank. As a prerequisite we find uniform formulas for the means and variances in terms of Coxeter group data. The main gadget for central limit theorems is the Lindeberg—Feller theorem for triangular arrays. Transferring the Lindeberg condition to the combinatorial setting, one finds that the validity of a central limit theorem depends on the growth of the dihedral subgroups in the sequence.