I will suggest two chapters in
Parthasarathy, Kalyanapuram Rangachari. Probability measures on metric spaces. Vol. 352. American Mathematical Soc., 1967.
primarily, but I guess I can add a few more if you can clarify my comment. If you are asking for the asymptotic distribution of such a statistics, then as long as a version of Central limit theorem(ergodic theorem) is applicable on the Lie group, then the resulting statistics will be $F$-free.