I am doing a modeling study using q-exponential distribution. I come to this question that are q-exponential distributions closed under convolution and multiplication operations? Let say X1 and X2 are independent q-exponential distributions, are X=X1+X2 and Y=X1*X2 also q-exponential?
I doubt this would happen, so I came across with two different methods to handle this, however, lacking a deep probability knowledge, I am not sure how much any of these approaches are valid or stable. Since I am going to perform the approximations in iterative manner and many times.
1- To approximate each of the q-exponentials by the respective Exponential distributions and perform the convolution operation on the Exponentials, next transform the resulting Exponential to a q-exponential distribution.
2- To take the first three moments of X1 +X2 or X1*X2 and then determine the approximation q-exponetial according to these moments.
I would appreciate it if you can help me in this regards.