Suppose a random variable $Z_n=(X_n,Y_n)$ takes a value on $A= [0,1] \times B=[0,1]$. We can consider a conditional distribution of $Y_n$ given the realized value of $X_n$. Now suppose $Z_n$ converges to $Z$ in distribution. What can I say about the convergence of the conditional distribution $F_n(Y_n|X_n=x)$? Can I say, for example, for $x$ almost surely, $F_n(|X_n=x)$ converges to $F(|X=x)$ in distribution?
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