New answers tagged convergence
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$X_n$ converges to $X$ in distribution. $Y_n$ converges to $Y$ in probability. Does $(X_n, Y_n)$ converges to $(X,Y)$ in distribution?
While this is wrong in general as per Zhanxiong's answer, it holds true if $Y$ is almost surely constant. I.e.
If $X_n \overset{d}\to X$ and $Y_n \overset{p}\to c$, then $(X_n, Y_n)\overset{d}\to (X, ...
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