Distribution/Variance of correlated squared normal random variables

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!

• Rotate to new coordinates which are uncorrelated. – kimchi lover Apr 13 at 17:11
• Thanks! How do I do that? – CLBJ_23 Apr 13 at 20:25
• Your covariance matrix is a multiple of the diagonal matrix plus a rank 1 matrix, so it has $(N-1)$ eigenvalues that are equal plus one more, corresponding to the eigenvector of all $1$s. From which you can deduce that $Z$ is the sum of a multiple of a degree-of-freedom $N-1$ chi squared rv plus a different multiple of a degree-of-freedom $1$ chi square. – kimchi lover Apr 13 at 22:32