Solved – Distribution of difference of two random variables with chi-squared distribution

Supose that we have two random variables $X sim chi_k^2$ and $Y sim chi_k^2$, with the same degrees of freedom.

A chi-squared distribution cannot have zero degrees of freedom, so what would be the distribution of $X – Y$ ?

This is not a chi-squared density, $X-Y$ will have support on $(-infty, +infty)$.

If the two variables are independent, it has mean 0 and variance $4k$.

If $k$ is large, its density is well approximated a normal variable with 0 mean and $4k$ variance. In the general case, its MGF has a closed form: $$E(exp(t(X-Y))) = (1-4t^2)^{-k/2}$$

If the 2 variables have dependence, the nature of that dependence needs to be explicited.

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