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 \(E\left(X_{n}^{3}ight)<\infty\). Suppose that \(F\) is a non-lattice distribution, then

\[\begin{equation*}F_{n}(x)=\Phi(x)+n^{-1 / 2} r_{1}(x) \phi(x)+n^{-1} r_{2}(x) \phi(x)+o\left(n^{-1}ight), \tag{7.58}\end{equation*}\]

as \(n ightarrow \infty\) uniformly in \(x\). Prove that \(q_{1}(x)=-r_{1}(x)\) and

\[q_{2}(x)=r_{1}(x) r_{1}^{\prime}(x)-\frac{1}{2} x r_{1}^{2}(x)-r_{2}(x)\]

by inverting the expansion given in Equation (7.58).