Density of a quadratic transformation of a normal random variable Consider the normally distributed random vector 
$$X \sim \mathcal{N}(\mu, \Sigma)$$
What is the distribution of $Y = f(X)$? 
For general $f$ this is a challenging problem but for the affine linear case 
$$f(x)_i = c_i + L_{ij}x_j$$
with $c$ a vector and  $L$ a matrix. We know that this has a nice closed form. In fact, $Y$ is distributed again as a multivariate normal. 
$$Y \sim \mathcal{N}(c + L\mu, \;L\Sigma L^T)$$
Consider now the next simplest case. Consider that $f$ is not linear but rather quadratic. I.e. 
$$f(x)_i = c_i + L_{ij}x_j + H_{ijk} x_jx_k$$
with $c$ a vector, $L$ a matrix and $H$ a rank three tensor. 
Does a closed form expression exist for the density of $Y$?
 A: The moments of such transformation can probably be found in Sec. 2.2.3 of Kollo and von Rosen (2005). Transformations of this kind have been used in some multivariate simulations. I understand there's a book on polynomials of multivariate distributions, but I have not seen it, and don't know if you'd be able to find the closed form expressions for the density of this transformation there. In a univariate case, you get a (scaled and shifted) non-central $\chi^2$ distribution, and the density expression for it is somewhat unwieldy (Bessel and hypergeometric functions, or infinite series of Poisson-weighted gamma distributions).
A: A related question was answered here. For that question the focus was on a particular quadratic form (the squared Euclidean norm), but this must also be one of the first things to consider. 
A recommended reference is the book "Quadratic forms in random variables" by Mathai and Provost (1992, Marcel Dekker, Inc.), but I can't remember exactly how general it gets, I don't have the book, and there is no Google preview. 
The results for the norm with general $\mu$ and $\Sigma$ clearly suggest that there is no simple closed form expression for the density. 
