This table is yet unfinished.

If I observe the sum of two processes with known distributions, the distribution of the observations is expected to be …

+ | Normal | Poisson | Binomial | Uniform | |

Normal | Normal | See here | |||

Poisson | Poisson, normal if many summands |
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Binomial | Binomial, if common p, Poisson, if many summands, Poisson binomial otherwise, note also the binomial sum variance inequality and this. |
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Uniform | Irwin-Hall |

Wikipedia has some more. A general discussion of probability distribution convolutions is for example here.