Combination of two Central Limit Theorems

Question

Suppose $\{X_n\}$ and $\{Y_n\}$ are independent sequences of postive valued random variables. Furthermore, assume there exist constants $\mu_1$, $\mu_2$, $\sigma_1$ and $\sigma_2$, such that \begin{equation} \frac{X_N-N\mu_1}{\sqrt{N}\sigma_1}\rightarrow \mathcal{N}(0,1)\text{ and }\frac{Y_N-N\mu_2}{\sqrt{N}\sigma_1}\rightarrow \mathcal{N}(0,1)\text{, as $N\rightarrow \infty$.} \end{equation} Can it be concluded that \begin{equation} \frac{Y_{X_N}-X_N\mu_1}{\sqrt{X_N}\sigma_1} \rightarrow \mathcal{N}(0,1)\text{, as $N\rightarrow \infty$}? \end{equation}

Answer 1

Assuming that every $X_N$ is positive integer valued (otherwise $Y_{X_N}$ makes little sense), note that as soon as $X_N\to\infty$ in probability and the CLT holds for $(Y_n)$, then the CLT for $(Y_{X_N})$ holds as well.

Thus, the answer to your question is "yes" when $\mu_1\gt0$. That $X_N\to\infty$ in probability means that $P[X_N\leqslant x]\to0$ when $N\to\infty$, for every finite $x$ hence a full CLT for $(X_N)$ is not necessary.