most probable value of estimator of k-th central moment

Hi all, I have a (possibly) naive question, hope someone can give me
some
references. Consider the estimator of the k-th central moment from N
i.i.d. data:
mu_k(N) = 1/N \Sum_{i=1,N} (X_i)^k
(let's assume X_i has zero mean). Is there any general result for the
N-dependence
of the pdf of mu_k, for example if the X_i are normal distributed?
Specifically
I'm interested in the most probable value of mu_k(N).
Not asking for exact results, bounds would be good...
thanks.

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