Let $X\sim\mathcal{N}_d(0,\Sigma)$ be a $d$-dimensional gaussian vector, where $0\in\mathbb{R}^{d}$ and $\Sigma\in\mathbb{R}^{d\times d}$ is diagonal. I'm interested on the distribution of: $$ ||X||^2 $$ I know that it follows a generalized chi-squared distribution , in general of the form $\tilde{\chi}^2(w,k,\lambda,m,s)$, where I used the notations of:
https://en.wikipedia.org/wiki/Generalized_chi-squared_distribution
In my particular case I think that $w = diag(\Sigma)$ (the vector whose entries are the diagonal's entries of $\Sigma$), $k = (1,1,\dots,1)$ , $\lambda = (0,0,\dots,0)$ , $m = s = 0$.
I would like to know how to prove this fact since I'm kind of "new" about the generalized chi squared distribution. Thank you in advance for your help.