Convergence in distribution of the sum of two dependent random variables

Question

I have the following question about the limiting distribution of the sum of two random variables say $Z_n = X_n+Y_n.$ I know the following:

• Conditioned on $X_n,$ $Y_n$ has a CLT i.e.,

$$\mathbb P (Y_n \le z | X_n) \to \phi(z)$$

where $\phi(z)$ is the cdf of a standard gaussian independent of $X_n.$

• Also, $$\mathbb P (X_n \le z) \to \phi(z)$$

From these two facts can I conclude $Z_n$ converges to $\mathcal{N}(0,2)$ in distribution?

Answer 1

Use characteristic functions. $Ee^{it(X_n+Y_n)} =E e^{itX_n}E(e^{itY_n}|X_n)$. Note that $E(e^{itY_n}|X_n) \to \phi (t)$ uniformly and $E e^{it(X_n)} \to \phi (t)$. It follows easily from these that $Ee^{it(X_n+Y_n)} \to \phi (t)^{2}$.