Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\) with mean \(\theta\) and variance \(\sigma^{2}\). Suppose that \(f\) has a finite fourth moment and let \(g\) be a function that has at least four derivatives. If the fourth derivative of \(g^{2}\) is bounded then prove that

\[V\left[g\left(\bar{X}_{n}ight)ight]=n^{-1} \sigma^{2}\left[g^{\prime}(\theta)ight]^{2}+O\left(n^{-1}ight)\]

as \(n ightarrow \infty\).