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I'm trying to sort out two different uses of the term "compound distribution" and figure out the relationship.

The Wikipedia article on compound distribution -- which I wrote -- defines a compound distribution as an infinite mixture, i.e. if $p(x|a)$ is a distribution of type F, and $p(a|b)$ is a distribution of type G, then $p(x|b) = \int_a p(x|a) p(a|b) da$ is a compound distribution that results from compounding F with G. This is the distribution of prior and posterior predictive distributions in Bayesian statistics.

However, the term "compound distribution" has another meaning as a random sum, i.e. a sum of i.i.d. variables where the number of variables is random.

What's the relation between the two? And am I using "compound distribution" correctly for the first definition?

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I can't say in general, but in the actuarial literature, the random sum of random variables is called an aggregate distribution, as in aggregate insured losses. Your definition of compound distribution is the one used in insurance.

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Check Feller V1

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