$\overline{X}\overset{p}{\to} 0$, $\overline{Y}\overset{p}{\to} 1$ where $\overline{X}=\frac{1}{N}\sum X_i$ and $\overline{Y}=\frac{1}{N}\sum Y_i$. Also $X_i\sim iid$ , $Y_i \sim iid$ but $X_i \not\perp Y_i$.
If I define $Z_i=\frac{X_i}{Y_i}$ what CLT can I apply to get some distribution of $\sqrt{n} \overline{Z}$ in terms of distribution of $\sqrt{n}\overline{X}$
Some more description:
Think of $X_i=a_i/c_i$ while $Y_i=b_i/c_i$. Individually $\{a_i\}$ and $\{b_i\}$ are not stationary. But $\{a_i/c_i\}$ and similarly $\{b_i/c_i\}$ are stationary. In fact $X_i=a_i/c_i \sim (0,\sigma^2)$ and $\mathbb{E}(Y_i)=1$. So I define $Z_i=X_i/Y_i$ as I am interested in $a_i/b_i$. From observed data series $\{a_i/b_i\}$ I am trying to come up with an estimate of $\sigma^2$. $\{c_i\}$ is not observable.
My guess is $\sqrt{N}\frac{1}{N}\sum \frac{X_i}{Y_i}\overset{d}{\to} (0,\sigma^2)$ but I may be wrong.