The n-fold convolution of a finite mixture of densities

In this paper, the probability density function of the n-fold convolution of a finite mixture of densities is obtained. The new density is again a finite mixture of densities. In this way, the formula recently given by Ma [N.-Y. Ma, A comment on “On the distribution of Ma and King, Applied Mathematics and Computation” 218 (2011) 202–203] for a two-fold convolution is interpreted and extended and a correct expression for the formula provided by Nadarajah [S. Nadarajah, On the distribution of Ma and King, Applied Mathematics and Computation 189 (2007) 732–733] is given. Several relevant examples are provided, including the convolution for the generalized exponential-sum distribution.