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 $Y$.

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 $Y$ ?