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.

  • $\begingroup$ way to go with delta method. $\endgroup$ – Khashaa Dec 14 '14 at 8:27

As given in A. Dasgupta (2008) Asymptotic Theory of Statistics and Probability, ch 3, p. 42,

enter image description here

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.