Does convergence in mean imply convergence in mutual information?

Question

Suppose $X_n (n \leq 1)$ and $Y_n (n \leq 1)$ be two sequences of random variables, if $X_n \rightarrow X$ and $Y_n \rightarrow Y$ in mean, does it imply that $I(X_n,Y_n)\rightarrow I(X,Y)$?

By $I(X, Y)$, I mean mutual information of $X$ and $Y$.

Answer 1

Not necessarily. Consider for a random variable $A$ having finite mean, $X_n = A/n$, $Y_n = A/n$. Then $X_n$ and $Y_n$ converge in mean to zero, but $I(X_n,Y_n) = I(A,A) \nrightarrow I(0,0)$.