  We consider generalized inversions and descents in finite Weyl groups. We
establish Coxeter-theoretic properties of indicator random variables of
positive roots such as the covariance of two such indicator random variables.
We then compute the variances of generalized inversions and descents in
classical types. We finally use the dependency graph method to prove central
limit theorems for general antichains in root posets and in particular for
generalized descents, and then for generalized inversions.
