$X_n\to \textrm{Exp}(5)$ and $Y_n\to 10$ in distribution. To what does $E[X_n Y_n]$ converge?

Question

I am solving the following problem.

$X_n\to \textrm{Exp}(5)$ and $Y_n\to 10$ both in distribution. To what does $E[X_n Y_n]$ converge?

My attempt:

From Wikipedia, I know that $(X_n, Y_n)\to (X, 10)$ in distribution, and also that $X_n Y_n \to 10X$ by combining the fact that “convergence to a constant in distribution implies convergence in probability” and Slutsky. However, I believe convergence in distribution does not say anything about the convergence of mean. So I do not know how to proceed.

Answer 1

It doesn't necessarily need to converge, and can converge to anything you like. Let $Y_n = 10 + Z_n$ where $Z_n$ is independent of $X_n,X$ and is some classic counterexample where $Z_n\to_p0$ but $E(Z_n)$ converges to something nonzero, or goes to infinity (e.g. let it be an independent sequence that is $n^2$ with probability $1/n).$