Prove Theorem 7.8. That is, let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\). Let \(F_{n}(t)=P\left[n^{1 / 2} \sigma^{-1}\left(\bar{X}_{n}-\muight) \leq tight]\) and assume that \(E\left(X_{n}^{3}ight)

\[F_{n}(x)-\Phi(x)-\frac{1}{6} \sigma^{-3} n^{-1 / 2} \mu_{3}\left(1-x^{2}ight) \phi(t)=o\left(n^{-1 / 2}ight),\]

as \(n ightarrow \infty\) uniformly in \(x\). The first part of this proof is provided after Theorem 7.4. At what point is it important that the distribution be nonlattice?

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Theorem 7.8. Let {X} be a sequence of independent and identically dis- tributed random variables from a distribution F. Let Fn(x) = P[n(Xn-