I am looking for the CDF of the difference of two dependent random variables : $Z=X-Y$ such that :
\begin{equation} X = \ \left| \sum_{\ell=1}^L\alpha_\ell C_\ell \right| ^2 \end{equation}
\begin{equation} Y = \max_k \left| \sum_{\ell=1}^L \alpha_\ell D_{\ell k} \right| ^2 \end{equation}
where: $ \alpha_\ell $ is a Gaussian complex random variable $\alpha_\ell\sim \mathcal{N}(0,P_\ell)$. $ C_\ell $ and $ D_{\ell k} $ are deterministic complex numbers, $k={1,1,\ldots,K}$.
So far, I found that $ X \sim \operatorname{Exp}\left(\frac{1}{\sum_{\ell=1}^L P_\ell \vert C_\ell \vert^2}\right)$ and the CDF of $Y$ as $ F_Y(y)= \prod _{k=1}^K (1- \exp(-\lambda_k y)) $ with $\lambda_k$ being a real deterministic parameter.
How can I proceed to find the CDF of $X-Y$ knowing that $X$ and $Y$ are not independent and I don't know how to get their joint pdf?