# Slutsky for joint convergence

I am interested whether Slutsky's Theorem also holds in the case of joint convergence. Let $(X_n,Y_n)$ be random variables with $(X_n,Y_n) \rightarrow (X,Y)$ in distribution for $n \to \infty$. Furthermore, let $(a_n,b_n) \rightarrow (1,1)$ a.s. for $n \to \infty$. Does also $(X_n*a_n,Y_n*b_n) \rightarrow (X,Y)$ hold in distribution for $n \to \infty$?

Proof 1: By Skorohod's representation theorem, there exists a probability space and random variables $X_n',Y_n'$, $X',Y'$ on this probability space with $$(X_n,Y_n) \sim (X_n',Y_n'), \qquad (X,Y) \sim (X',Y')$$ such that $$(X_n',Y_n') \to (X',Y') \qquad \text{almost surely.}$$ Since $a_n \to 1$ and $b_n \to 1$, we get $(a_n X_n', b_n Y_n') \to (X',Y')$ almost surely. Now, as $(a_n X_n',b_n Y_n') \sim (a_n X_n, b_n Y_n)$, it follows that $(a_n X_n,b_n Y_n) \to (X,Y)$ in distribution.