Suppose that $a \in \mathbb{R}^n$ has each of its entries being i.i.d. random variables drawn from a Gaussian distribution with mean zero and variance 1, i.e. $a_i \sim N(0,1), i=1, \cdots, n.$
I am working on calculating the expectation value $\mathbb{E}[\|a\|_2^{2m}]$, where $m$ is a positive integer.
It seems that one strategy might be to expand $\|a\|_2^{2m}$ using the multinomial distribution, and then apply the linearity of the expectation to evaluate each of the terms in the summation. However, keeping track of the expectation of each of these terms still appears unwieldy.
I would be very grateful for any assitance in evaluating this expectation. Thank you very much.
Note that the probability density function depends purely on the radial component $$ f_R(r)=\int_{S^{n-1}(r)}f(\mathbf{x})\,\mathrm{d}\mathcal{H}^{n-1}\propto r^{n-1}\exp(-r^2/2) $$ Hence you can immediately read off $\mathbb{E}(\|a\|_2^{2m})$ as a quotient of two Gamma: $$ \begin{align*} \mathbb{E}(\|a\|_2^{2m})&=\mathbb{E}(R^{2m})\\ &=\frac{\displaystyle\int_0^\infty r^{2m}\cdot r^{n-1}\exp(-r^2/2)\,\mathrm{d}r}{\displaystyle\int_0^\infty r^{n-1}\exp(-r^2/2)\,\mathrm{d}r}\\ &=2^m\frac{\Gamma(m+\frac{n}2)}{\Gamma(\frac{n}2)}\\ &=n(n+2)\dots(n+2m-2). \end{align*} $$