Let $X_n$ and $Y_n$ converge in law to $X$ and $y$ respectively, where $y$ is a constant. Then are the following true?
- $X_nY_n \to yX$
- $X_n + Y_n \to X + y$
- $X_n/Y_n \to X/y$ (if $y\neq 0$)
These statements are obviously true for almost sure convergence, but what about convergence in probability? For convergence in distribution? In the convergence in distribution case I think we may have to suppose $Y_n$ converges to a constant.
I'd also be interested in references for these statements, because I can't find anything online.