Random Variable for which sum is exponentially distributed.

Question

Let $X_1,...,X_k\sim D$ IID. What can we say about distributions $D$ so $\sum_iX_i\sim E(\lambda)$? Do such distributions even exist?

What if $X_1,...,X_k$ are not IID?

Answer 1

Yes, it is easy to see that if $$D \sim \operatorname{Gamma}(1/k, \lambda),$$ then $$\sum_{i=1}^k X_i \sim \operatorname{Gamma}(1, \lambda) \sim \operatorname{Exponential}(\lambda).$$ This establishes existence but does not say anything about whether such distributions exist for the non-IID case or characterize all such distributions.