I'd be happy for simply a reference or even search terms as I feel like this has to be known*.    

Suppose we have a known probability distribution XX and a fixed integer nn. I am interested in the following distribution on all the set of partitions of XX interest into n (non-empty, if need be) sets. If P=⨿XiP=⨿Xi is such a partition and mi=E(Xi)mi=E(Xi), then the distribution's value at PP is XP=∑minXP=∑min.

Is there something like the Central Limit Theorem for this case? If I take a random partition PP, I'd like to know something like its p-value - what do we expect for PP and how much variation can we expect?

This seems to be a way to define or test whether or not a division of XX into groups is meaningful: it is meaningful when the XPXP differs from a generic grouping by some fixed amount. That's why I would think this is known.

Thanks!

I first asked this question yesterday on mathstackexchange [here][1] and didn't think it was appropriate since it's not a research related question.  However,  the suggested question/answers to this on mathoverflow and the little attention I have received on mathstatckexchange suggests this might be a more appropriate site.  In either case, I will remove one of the two based on feedback.

  [1]: https://math.stackexchange.com/questions/2518928/central-limit-like-theorem-for-the-distribution-of-all-possible-partitions