We discuss the CLT for the global and local linear statistics of random matrices from classical compact groups. The mainparts of our proofs are certain combinatorial identities, much in the spirit of works by M. Kac and H. Spohn.
"The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities." Ann. Probab. 28 (3) 1353 - 1370, July 2000. https://doi.org/10.1214/aop/1019160338