Suppose $\{(X_i, Y_i)\}_{i=1}^\infty$ is an iid sequence of random variables. Assume $EX_i^2$ and $EY_i^2$ are both finite and that $E(X_i) \neq 0$. Define
$$Z_n = \frac{n^{-1}\sum_{i=1}^n Y_i}{n^{-1}\sum_{i=1}^n X_i}$$
I want to show
$$E(Z_n) \to \frac{E(Y_i)}{E(X_i)}$$
but haven't been able to do so. Here are some things I tried:
The result will follow from Jenson's Inequality if I can show $$Z_n \stackrel{L^1}{\to} \frac{E(Y_i)}{E(X_i)}$$ From the Law of Large Numbers and Continuous Mapping Theorem, I know that $$Z_n \stackrel{p}{\to} \frac{E(Y_i)}{E(X_i)}$$ I also know that convergence in probability implies $L^1$ convergence if $Z_n$ is uniformly integrable. I tried showing uniform integrability via the Cauchy-Schwartz Inequality, but had trouble bounding the expectation of $1 / \bar{X}_n^2$
I also can use the Delta Method to derive the limiting distribution of $Z_n$: $$\sqrt{n}\left(Z_n - \frac{E(Y_i)}{E(X_i)}\right) \stackrel{d}{\to}N(0, \sigma^2)$$ where $\sigma^2$ depends on the means and covariance of $(X_i, Y_i)$. I tried applying the Portmanteau Lemma, but couldn't figure out how to use bounded functions to get the result.
I also know that $$\bar X_n \stackrel{L^2}{\to} E(X_i), \qquad \bar Y_n \stackrel{L^2}{\to} E(Y_i)$$ from the weak Law of Large Numbers. But since the function of interest is not Lipschitz, I could not find a continuous mapping theorem to apply.
No. The expectation of $Z_n$ may not even exist. Say, let $X_i$ are normal $\mathcal N(1,1)$ and do not depend on $Y_i$, then $S_n=\sum_{i=1}^n X_i \sim N(n,n)$ and $$ \mathbb E\left[\frac1{|S_n|}\right] = \int_{-\infty}^\infty \frac{1}{|x|\sqrt{2\pi n}} e^{-\frac{(x-n)^2}{2n}}dx = +\infty $$ Integral diverges at zero for any $n$.