Prove Theorem 6.4. That is, suppose that \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(\sigma_{n}^{-1}\left(X_{n}-\muight) \xrightarrow{d} Z\) as \(n ightarrow \infty\) where \(Z\) is a \(\mathrm{N}(0,1)\) random variable and \(\left\{\sigma_{n}ight\}_{n=1}^{\infty}\) is a real sequence such that

\[\lim _{n ightarrow \infty} \sigma_{n}=0\]

Let \(g\) be a real function such that \[\left.\frac{d^{m}}{d x^{m}} g(x)ight|_{x=\mu} eq 0\]
for some \(m \in \mathbb{N}\) where \[\left.\frac{d^{k}}{d x^{k}} g(x)ight|_{x=\mu}=0\]
for all \(k \in\{1, \ldots, m-1\}\). Then prove that \[\frac{m !\left[g\left(X_{n}ight)-g(\mu)ight]}{\sigma_{n}^{m} g^{\prime}(\mu)} \xrightarrow{d} Z^{m}\]
as \(n ightarrow \infty\).

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Theorem 6.4. Let {X} be a sequence of random variables such that (Xn-) Z as n where Z is a N(0, 1) random variable and {n} is a real sequence such that lim On = = 0. Let g be a real function such that dm dxm9(x) # 0, x=l