Let $Y$ be a symmetric random variable, $(X\_n)\_n$ be a sequence of nonnegative random variables, and set $p\_n = \mathbb P(Y \le X\_n)$. It is known from Slutsky's theorem that, if $c$ is a constant such that $X\_n \to c$ in probability, then $p\_n \to F\_Y(c)$, where $Y$ is the CDF of $Y$.



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> **Question.** Can convergence of $X\_n$ in probability to a constant $c$, be replace by some other notion of convergence, say $X\_n \to X$ (in some sense), for some random variable $X$, such that we can still compute the limit of $p\_n$ only via the data $X$ and $Y$ ?
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Compute limit of $\mathbb P(Y \le X_n)$ using limiting information on the sequence of random variables $X_n$
Even if $X\_n\to c$ in probability for some real constant $c$, it is not necessary that $P(Y\le X\_n)\to P(Y\le c)$ -- you also need to require that $P(Y=c)=0$.


More generally, if the limit of $X\_n$ is not a constant, then you need to assume the convergence, not just of the distribution of $X\_n$, but of the joint distribution of $X\_n$ and $Y$. In particular, if $(X\_n,Y)$ converges to $(X,Y)$ in distribution and $P(Y=X)=0$, then, by the [Portmanteau theorem](https://en.wikipedia.org/wiki/Convergence_of_measures#Weak_convergence_of_measures), $P(Y\le X\_n)\to P(Y\le X)$.


