Expected value of power of sum of square of dependent Gaussian

So, I have a set of dependent Gaussian RVs $\{X_k\}_{k=1}^{N}$ with known joint PDF (zero mean and given covariance matrix). I'm interested in whether we can compute the quantity: $$\mathbb{E}\left[\left(\sum_{k=1}^{N}{X_k^2}\right)^{\beta}\right],$$ where $\beta > 0$.

I tried thinking about the sum and see if I can linked to Chi-squared distribution, but the problem is that the set of variables in question are dependent.

• math.stackexchange.com/questions/442472/… This should be related. – BGM Sep 10 '18 at 5:37
• Thanks for directing me to that link. The thing is that the power here is making the problem harder or at least this is what I'm thinking. – Jeremy Sep 10 '18 at 5:43

Is $\beta$ an integer or what?
If what you actually know is that the vector $\vec X=(X_1,\ldots,X_N)$ follows a multivariate normal distribution with zero mean and covariance matrix $\Sigma$, then you have joint density $$f_{\vec X}(\vec x)=\frac1{\sqrt{2\pi \det(\Sigma)}}e^{-\frac12\vec x^T\Sigma^{-1}\vec x},$$ where $\vec x=(x_1,\ldots,x_N)^T$.
So, as a very general answer, we can say that $$E\left[\left(\sum_{k=1}^N X_k^2\right)^\beta\right]=\int_{\mathbb R^N}\left(\sum_{k=1}^N x_k^2\right)^\beta \cdot f_{\vec X}(\vec x)\quad dx_1\ldots dx_N,$$ but I can't think of a general expression for this integral.
• $\beta$ is a real number. I tried writing down what you have written by I fail to see how can I possibly move forward. Would it change something if $\beta$ was an integer? using binomial formula? – Jeremy Sep 10 '18 at 6:48