Asymptotic probability distributions of some permutation statistics for the wreath product \(C_r \wr S_n \)

We present in this work results on some distributions of permutation statistics of random elements of the wreath product \( G_{r,n} = C_r \wr S_n \). 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.