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Let $\mathcal{D}_1$, $\mathcal{D}_2$ and $\mathcal{D}_3$ be three distributions defined on some space $\mathcal{X}$.

Assume $\mathcal{D}_1 = \mathcal{D}_2 + \mathcal{D}_3$

and $\mathcal{D}_1$ is assumed to be known and for example it's a (multi) variate normal distribution, what can we say about $\mathcal{D}_2$ and $\mathcal{D}_3$, are they also gaussian? The problem has many degrees of freedom but I just want to get some intuition about what those distributions could be.

WHat happen if we generalize the problem to other classical laws and to exponential families ?

rivana
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    When you write $\mathcal{D}_2+\mathcal{D}_3$, can you tell us what you mean by the sum of two distributions? – John Madden Dec 02 '23 at 17:22
  • Look at the characteristic functions and apply https://stats.stackexchange.com/questions/512568. – whuber Dec 03 '23 at 00:28

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