# Convergence in distribution of the sum of two dependent random variables

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?

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 $$Ee^{it(X_n+Y_n)} \to \phi (t)^{2}$$.