1. ## Convergence in Distribution

Question
Prove whether this is true or false:

$Y_n \rightarrow^{d}Y$ and $X_n \rightarrow^{d}X$ $\Rightarrow Y_n-X_n \rightarrow^{d}Y-X$

I don't know how to do this! Can we simply use the continuous mapping theorem?? Please help!

Thank you!

2. what happens if $X_n=Y_n$ and let $X_n$ converge to a N(0,1).

3. Originally Posted by matheagle
what happens if $X_n=Y_n$ and let $X_n$ converge to a N(0,1).
Sorry, I don't follow.

If $X_n \rightarrow N(0,1)$ and $X_n = Y_n$ then $Y_n \rightarrow N(0,1)$

So $Y_n - X_n \rightarrow 0$ and it works??

Sorry, can you please expand a bit more? Thanks.