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.