Converse Version of Slutsky's Theorem

Question

Let $X_n$ and $Y_n$ be two random sequences. According to Slutsky's theorem, we know that if $X_n \overset{P}{\to} c$ for $c \in \mathbb{R}^1$ and $Y_n \overset{d}{\to} Y$, then $X_nY_n \overset{d}{\to} cY$. I am wondering a converse version of this. That is, if $X_nY_n \overset{d}{\to} c Y$ and $X_n \overset{P}{\to} c$, is it true that we have $Y_n \overset{d}{\to} Z$ for some random variable $Z$ (possibly $Z=Y$)?

Many Thanks.

Answer 1

This fails if $c=0$ but holds otherwise: You can define $\widetilde{Y_n} =X_n Y_n$ and $$\widetilde{X_n} =1_{X_n=0}+1_{X_n\ne 0} \cdot X_n^{-1} \,, $$ and apply Slutzky's theorem to these new variables. Note that if $X_n \overset{P}{\to} c \ne 0$ then $\widetilde{X_n} \overset{P}{\to} 1/c$.