An (possible) application of the Central Limit Theorem

Question

Consider $(z_i)_{i=1}^n$ a sequence of i.i.d random variables with $\mu \neq 0$ and variance $\sigma^2<\infty$. Let $\bar{z_i}$ be the sample mean. Show that:

$$\sqrt{n}\left(\frac{1}{\bar{z_i}}-\frac{1}{\mu}\right)\overset{d}{\to}N\left(0,\frac{\sigma^2}{\mu^2}\right)$$

So far, what I did was trying to definde a new variable $y_i=\frac{1}{z_i}$.I have been trying to compute $Var(y_i)$ in order to use the Central Limit Theorem. However I have been obtaining the following:

$$Var(y_i)=\mathbb{E}y_i^2-[\mathbb{Ey_i}]^2$$

$$=\frac{1}{\mathbb{E}z_i^2}-\frac{1}{\mu}$$

However, I obtain $\mathbb{E}z_i^2=\sigma^2=\mu^2$, which doesn't help obtain the $\sigma^2/\mu^4$ needed in the variance. Any suggestions? I sthere something I am doing wrong?

Answer 1

Thanks to snarfblaat for the suggestion. Recall that by the Delta Method, we have that:

$$\sqrt {n} (g(z_i)-g(\mu))\to N(0,\sigma^2[g'(\mu)]^2)$$

Now, just pick $g(x)=1/x$ and we get to the desired result.