Joint asymptotic normality

Question

Suppose I have three sequences of random variables, $X_n$, $Y_n$ and $Z_n$. I know $X_n$ and $Y_n$ are jointly asymptotically normal, i.e., $X_n \overset{d}{\to} X$, $Y_n \overset{d}{\to} Y$, $X$ and $Y$ are both normal, and jointly $X_n$ and $Y_n$ converges in distribution to a bivariate normal random vector. If I know $Z_n \overset{p}{\to} c$, a constant. Is this sufficient to claim that $X_n+Y_nZ_n$ is asymptotically normal?

Answer 1

Having thought about it a bit more myself ... I think it is true, because:

$$X_n+Y_nZ_n = X_n + cY_n + (Z_n-c)Y_n.$$

Now $X_n$ and $c Y_n$ are obviously jointly asymptotically normal by my assumption, hence $X_n+cY_n$ is asymptotically normal. And $(Z_n-c)Y_n$ converges in probability to $0$ by Slutsky's theorem. So this whole thing converges in distribution to something that's normal.