How to obtain the gamma distribution through convolution of two different distributions? How can one obtain the gamma distribution through convolution of two different distributions? Could the gamma distribution be created as a non-trivial sum of $N$ random variables $X$ which have the same distribution and parameters?
Trivial case is summation of fixed number $N$ of variables with gamma distribution as described on Wikipedia.
 A: Wikipedia answered the question for you.  The sum of any two independent  identically distributed gamma random variables is gamma.  So if you sum two independent identically distributed gammas you get a gamma.  Is that what you meant or did you want to sum two distributions that are identical but not gamma?  If that is the case I beleive the answer is no. Also since you mentioned N possibly greater than 2, it is also true as others have pointed out that any two gammas that are independent but not necessarily identically distributed will sum to a gamma if they have the same scale parameters. So you can get a gamma as the sum of n iid gammas. You see if you can get 2 by summing two iid gammas you can get 4 by adding another 2 iid gammas with the same distribution as the first two and so on.  The chi-square is a special case where this works.  Distributions like the normal and the chi square that can be represented as the sum of n iid random variables of the same form (normal or chi-square respectively) are called infinitely divisible.
A: You could also sum the squares of independent standard normal variates, giving chi-square random variables, which are a special case of the gamma.
