Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert\_2$. 


If $\Sigma = {I}\_N$, we recognize the $\chi$-law. We especially know that the probability density is given by 
$$p(x) \propto x^{{N} -1} \mathrm{e}^{-\frac{x^2}{2}} 1\_{x\geq 0} .$$


In the general case, we can decompose the matrix $\Sigma = P^t D P$ with $P$ orthogonal and $D=D(\lambda\_1,\cdots,\lambda\_N)$ diagonal and $\lVert X \lVert\_2 \sim \lVert \mathcal{N} (0,D)\lVert\_2$. What can we say about the probability density of $\lVert X \lVert\_2$ in this general case? 


Thanks by advance. 



Euclidian norm of Gaussian vectors
After you diagonalize the covariance matrix, you have $\|X\|^2=\sum\_{k=1}^n X\_i^2$, right? And $X\_1^2,\ldots,X\_n^2$ are independent $\chi(1)$-distributed r.v. (up to a scaling with $\frac{1}{\sqrt{\lambda\_i}})$, i.e., they are gamma-distributed, but with different scale parameter. So you can apply the results of this paper:


P. G. Moschopoulos (1985) The distribution of the sum of independent gamma random variables, Annals of the Institute of Statistical Mathematics, 37, 541-544.


See also


A. M. Mathai (1982) Storage capacity of a dam with gamma type inputs, Annals of the Institute of Statistical Mathematics, 34, 591-597.


