Let $X_1$, $X_2$, $\ldots$ iid, $\sim$ $N(\theta,\sigma^2)$; $\theta \in \mathbb{R}$ unknown, $\sigma > 0$ and $a \in \mathbb{R}$ given. $g(\theta) := P(X_i > a)$.
We want to estimate $g(\theta)$ using $[X_1, X_2, \ldots , X_n]^{\prime}$ with $\hat{g}_n := \Phi \left( \frac{\bar{X_n}-a}{\sigma} \right)$ where $\Phi$ is the distribution function of $N(0,1)$.
I have to show that $\sqrt{n}\left( \hat{g}_n - g(\theta) \right)^2 \overset{d}{\rightarrow} N(0,v^2)$ and determine $v > 0$.
But I don't understand how I have to modify the delta-method that it works for $()^2$ or maybe there is another way to show this.