On Wikpedia and in the references therein [1,2], it is stated that any quadratic form of normally distributed random variables can be expressed as the sum of many independent non-central chi-squared distributed variables and a single normally distributed variable.

It is not obvious to me how to prove that, and there is no reference to an actual proof as far as I can see.

Let's be more precise. I'm interested in the random variable

$$X = {\mathbf z}^T A \mathbf{z} + B^T \mathbf{z},$$

where $\mathbf{z}$ is a random vector with zero mean and identity covariance, $A$ is a matrix (usually symmetric), and $B$ is a vector (I think any quadratic form can be reduced to this form with the appropriate transformations and ignoring deterministic constants. Correct me if I'm wrong.).

Then the problem is to show that $X$ can be written as

$$X = \sum_{i=1}^r \lambda_i Y_i + \mu Y_0,$$

where the $\lambda_i$ and $\mu$ are constants, the $Y_i$ follow non-central chi-squared distributions, and $Y_0$ follows a standard normal distribution.

Does someone know how to prove that statement? And it is possible to get explicit expressions for the coefficients $\lambda_i$ and $\mu$, as well as for the parameters of the non-central chi-squared distributions of the $Y_i$?

EDIT: I had previously missed it, but I think a very similar (if not identical) question is already addressed here.


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