When does $X_k\to\ N(\mu_1,\sigma^{2}_1)$ and $Y_k\to\ N(\mu_2,\sigma^{2}_2)$ imply $X_k+Y_k$ also converges to Normal Distribution?

Question

Let $\{X_1, X_2, …\}$ and $\{Y_1, Y_2, …\}$ be sequences of i.i.d. random variables (however, $X_i$s are not independent of $Y_i$s). Suppose that $X_k\ {\xrightarrow {d}}\ N\left(\mu_1,\sigma^{2}_1\right)$ and $Y_k\ {\xrightarrow {d}}\ N\left(\mu_2,\sigma^{2}_2\right)$

The question is, under which conditions we have that the sum $X_k+Y_k$ also converges to the normal distribution?

Answer 1

If $(X_n,Y_n)$ are jointly normal such that $(X_n,Y_n) \rightarrow (X,Y)$ where $X \sim N(\mu_1,\sigma_1^2)$ and $Y \sim N(\mu_2,\sigma_ 2^2)$, then $X_k + Y_k \xrightarrow{D} X + Y$. This is a consequence of the Continuous Mapping Theorem, applied to the function $g(x,y) = x + y$.