Asymptotic probability distributions of some permutation statistics for the wreath product $C_r wr mathfrak{S}_n$

We present in this work results on some distributions of permutation statistics of random elements of the wreath product $G_{r,n} = C_r wr mathfrak{S}_n$. We consider the distribution of the descent number, the flag major index, the excedance, and the number of fixed points, over the whole group $G_{r,n}$ or over the subclasses of derangements and involutions. We compute the mean, variance and moment generating function, and establish the asymptotic distributions of these statistics.