Suppose $(X_n,Y_n)$ is a sequence of bivariate random variables. Suppose there exists a bivariate random variable $(X,Y)$ such that $X_n\to X$ and $Y_n\to Y$ in distribution. Suppose further that for all $a,b\in \mathbb{R}$ we have $$\max\{aX_n, bY_n\} \stackrel{d}{\to} \max\{aX,bY\}.$$ Then does $(X_n,Y_n)\stackrel{d}{\to} (X,Y)$?
I am not sure if it is at all true. The question is pretty similar to the Cramer Wold device but with the max operator. Note that we may write $$aX_n+bY_n= \max\{aX_n, bY_n\}+\min\{aX_n, bY_n\}=\max\{aX_n, bY_n\}-\max\{-aX_n, -bY_n\}.$$ However, this does not quite help. Any thought or help is appreciated.
One of my colleagues solved this problem if $X_n,X$ and $Y_n,Y$ are assumed to be strictly positive. Then my assertion is true. The idea is to take logarithms. Indeed we can do the following elementary computation: \begin{align*} \mathbb{P}(\log X_n \le -\log a, \ \log Y_n \le -\log b) & = \mathbb{P}(\max\{\log (aX_n),\ \log (bY_n)\} \le 0) \\ & = \mathbb{P}(\max\{aX_n,\ bY_n\} \le 1) \\ & \to \mathbb{P}(\max\{aX,\ bY\} \le 1) \\ & = \mathbb{P}(\log X \le -\log a, \log Y \le -\log b) \end{align*}
Varying $a,b$ over $(0,\infty)$, we see that $(\log X_n,\log Y_n) \stackrel{d}{\to} (\log X, \log Y)$, or equivalently $(X_n,Y_n) \stackrel{d}{\to} (X,Y)$.