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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.

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1 Answer 1

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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.

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Effectively, the paper of Moschopoulos (and actually the other one too) gives a formula for the density of $\lVert X \lVert_2^2$ from what we can deduce another one for $\lVert X \lVert_2$. It is expressed as an infinite sum. Thanks a lot for these references. –  Goulifet Jan 30 at 15:20

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