I'm struggling with the following exercise:
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $X,Y$ be two real-valued random varaibles such that their corresponding cumulative distribution functions $F_X$ and $F_Y$ are continuous and strictly monotonically increasing. Further, we assume that $(X,Y)\sim(q_X^-(U),q_Y^-(U))$, where the random variable $U$ is uniformly distributed on $(0,1)$. Show that in this case $$ VaR_\lambda(X+Y)=VaR_\lambda(X)+VaR_\lambda(Y) $$
My idea: Since $F_X$ and $F_Y$ are continuous and strictly monotonically increasing, there exist continuous inverse functions $F_X^{-1}$ and $F_Y^{-1}$ of $F_X$ and $F_Y$ such that $VaR_\lambda(X)=F^{-1}_X(\lambda)$ and $VaR_\lambda(Y)=F^{-1}_Y(\lambda)$. My main problem is that I don't really know what this $(X,Y)\sim(q_X^-(U),q_Y^-(U))$ actually means (where $q_X^-(\lambda)$ is the lower $\lambda$-quantile) or how to deal with this. One thing I saw is that for a uniformly distributed $U$ on $(0,1)$, we have that $F(X)=\mathbb{P}(U\leq F(X))=\mathbb{P}(q(U)\leq x)$. I'm not sure if this is correct in this setting or helpful, but these are the only approaches I have so far.
Thank you for your help.
Here's a hint on top of my comment:
$$X+Y \sim q_X^-(U) + q_Y^-(U) = (q_X^- + q_Y^-)(U) \; .$$