I have a problem that it (might) be possible to describe as a quadratic of the form
$$\sum_{i} a x_{i}^{2}+ b y_{i}^{2}+cx_{i}y_{i}$$
where $x_{i}$ and $y_{i}$ are independent normal variables but not necessarily with zero mean.
I have read some of the material on the wikipedia page on such forms, and also Kotz & Johnson have a chapter in Continuous Univariate Distributions on such material both of which give a relatively straightforward formula for mean & variance.
However, I am looking to probe the nonnormality of the resulting distribution somehow. Obviously $b=c=0$ leads to a chi-squared distribution. I wonder if there is some systematic way to analyse the general case. I have seen some reference to the case of $a,b>0$ but $c=0$ approximated to chi-squared. I am curious about $c\lessgtr 0$, and especially when $c<0$, (so as to offset some of the variance of the other two terms), but have so far drawn a blank on any leads on references.
Would anyone happen to know if anyone has published on this or a related topic?
Many thanks.