Example of product of sequence of random variables that does not converge in distribution to the product of the limits

Question

One result of Slutsky's theorem is that when $X_{n}$ is a sequence of random variables converging to a random variable $X$ and $Y_{n}$ a sequence of random variables converging to a constant $c$ then it holds that in the limit $n \rightarrow \infty$ we have that $$X_{n}Y_{n}\ {\xrightarrow {d}}\ cX$$.

But apparently this doesn't hold if the sequence $Y_{n}$ converges to a non-constant random variable $Y$, i.e. generally it does not hold that $$X_{n}Y_{n}\ {\xrightarrow {d}}\ XY.$$

Does anybody have a good intuitive counter-example for this case?

Edit: Both $X_{n}$ and $Y_{n}$ converge in distribution to their respective limits.

Answer 1

I am assuming you are looking for a counterexample of $X_n \overset{d}{\to} X$, $Y_n \overset{d}{\to} Y$ but $X_nY_n \overset{d}{\not\to} XY$. If so, let $X, Y$ be i.i.d. $N(0, 1)$ random variable, then let $X_n \equiv Y_n = X$. Then the condition $X_n \overset{d}{\to} X$, $Y_n \overset{d}{\to} Y$ is clearly satisfied. However, $X_nY_n = X^2 \sim \chi^2_1$ obviously does not converge in distribution to $XY$. One way to see this is by noting $P[X^2 \leq 0] = 0 \not\to P[XY \leq 0] = \frac{1}{2}$.