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Is there a way to find the asymptotic distribution for the moments of Gaussian distribution?

More specifically, say you have $X_1, ..., X_n \sim N(\mu, \sigma^2)$. For a moment $m_{n, k} =\frac{1}{n} \sum_{i=1}^n (X_i - \bar{X}_n)^k$, we want to find an asymptotic distribution $G$, such that: $$ a_n ( m_{n,k} - \mu_k ) \rightarrow G $$ For some sequence of numbers $a_n$. Or maybe in a more general form: $$ a_n ( m_{n,k} - b_n ) \rightarrow G $$ For some sequence of numbers $a_n$ and $b_n$.

PS. I am not sure how hard/easy the problem is. If it is easy, any hints/pointers are appreciated.

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  • $\begingroup$ way to go with delta method. $\endgroup$ – Khashaa Dec 14 '14 at 8:27
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As given in A. Dasgupta (2008) Asymptotic Theory of Statistics and Probability, ch 3, p. 42,

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The reference to Serfling is Serfling R.J. (1980) Approximation Theorems of Mathematical Statistics.

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