Suppose that $X_n \stackrel{p}\rightarrow X$. Is it true that $\mathbb{P}(Y|X_n)$ converges in distribution to $\mathbb{P}(Y|X)$? I.e. $\mathbb{P}(Y\leq y |X_n) \rightarrow \mathbb{P}(Y\leq y |X)$
This is probably not true, but I am interested in maybe some weak conditions where it is true. For example, if we have $(x,y) \mapsto \mathbb{P}(Y \leq y|X=x)$ is a continuous function of $x$ for every $y$...
The best I can get: if we have $(x,y) \mapsto \mathbb{P}(Y \leq y|X=x)$ is a continuous function of $x$ for every $y$, then $\mathbb{P}(Y\leq y |X_n) \stackrel{p}{\rightarrow} \mathbb{P}(Y\leq y |X)$ by the continuous mapping theorem, but I am not sure convergence in probability of distributions makes sense...