If $X_{1}, X_{2}, \ldots, X_{N}$ are identically distributed normal random variables with mean $0$ and variance $\frac{(N+3)D\sigma^{2}}{N}$, then I want to calculate the distribution, or at least the variance of $Z = X_{1}^{2} + X_{2}^{2} + \ldots + X_{N}^{2}$.
I already found that $Cov(X_{i}, X_{j}) = \frac{\sigma^{2}D}{N}$, $\forall i,j \in \{1, 2, \ldots, N\}$ and that $E[X_{i}^{2}] = \frac{(N+3)D\sigma^{2}}{N}$, $Var(X_{i}^{2}) = \frac{(N+3)^{2}{D^{2}\sigma}^{4}}{N^{2}}$, $\forall i,j \in \{1, 2, \ldots, N\}$.
My problem is that they are correlated and I do not know how to compute the variance and the distribution of $Z$.
Any help would be highly considered!