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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?

Alexander Mathiasen
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    A distribution is called infinitely divisible if for each $k$ it is the distribution of the sum $k$ i.i.d random variables. There is a huge literature on such distributions. – Kavi Rama Murthy Oct 09 '20 at 23:21

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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.

heropup
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