Does the convolution property of Gamma require independence?

Question

I understand that when α is an integer, the Gamma(α,β) distribution is the distribution of the length of time you have to wait until a total of α events have occurred, where the time between each event follows an Exponential(β) distribution.

However, does this property require each exponential distribution to be independent?

Answer 1

Yes, of course, independence is required. For a simple counterxample, take the case where all the exponentials are actually the same variable.