I am looking to determine the distribution of $\max(X_i,Y_i)$ where
$X_i = |A_i|^2$
$Y_i = \frac{1}{2}|A_i - A_{i-1}|^2$
Here $A_i$ is a complex vector with normally distributed real and imaginary components with $\mu=0$ and $\sigma=1$.
Both $X_i$ and $Y_i$ follow a $\chi^2$ distribution with $k=2$ and if they were independent, one would expect that $P(\max(X_i,Y_i)\leq x) = F_{\chi^2}(x)^2$ where $F_{\chi^2}(x)^2$ is the CDF of a $\chi^2$ distribution with $k=2$. However, $X_i$ and $Y_i$ both depend on $A_i$, with $\operatorname{Cov}(X_i,Y_i)=2$.
How does one determine the distribution of the maximum of covariant variables? I assume it is dependent on the conditional probability $P(Y_i \leq x\ |\ X_i \leq x)$.
$A_i=C_ie^{i\theta_i}$ where $C_i\sim \chi_2$ and $\theta\sim U[0,2\pi)$. Then $$X= C_2^2\sim \mathcal{E}(\frac 1 2)$$ $$Y=\frac 1 2(C_1^2+C_2^2-2C_1C_2Re(e^{i(\theta_1-\theta_2)}))$$ so to find $P(Y\le a|X\le a)$ you'll need to solve a single quadratic equation and integrate.