# Asymptotics - Delta Method used in proof

How can I use the delta method to show that

$\sqrt n$ (1/$\bar Y_n$ - 1/$\mu)$ $\rightarrow$ N (0, $\sigma^2$/$\mu^4$) ?

We know that:

$\bar Y_n$ = $1/n \sum_{n=1}^n Y_i$

$\ Y_i$ is i.i.d; $E(\ Y_i )= \mu\ne 0$ and $V(\ Y_i)= \sigma^2 \gt$ 0

Set $f(X) = \frac{1}{X}$ (for $f\colon W\longrightarrow \mathbb R$). Then $Df(X) = -\frac{1}{X^2}$ ($Df$ denotes the derivative). The delta-method states that $$\sqrt n(f(X) - f(\mu)) \rightarrow N(0,\sigma^2Df(\mu)^2)$$ provided that $\sqrt n(X-\mu)\rightarrow N(0,\sigma^2)$. Now your job is to plug in the right value for $X$ and you are done ;-)
• It's not necessary because we are allowed to use the delta method (however, delta method requires that $\sqrt n(X-\mu)$ converges in distribution. If this is not given, you have to proof it first of course, but that's a new question) – Syd Amerikaner Jan 11 '17 at 14:13