Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a distribution \(F\) where \(E\left(\left|X_{1}ight|^{k}ight)<\infty\). Consider the sum

\[\sum_{j=1}^{k}\left(\begin{array}{l}k \\j\end{array}ight) E\left\{\left(\mu_{1}^{\prime}-\hat{\mu}_{1}^{\prime}ight)^{j}\left[n^{-1} \sum_{i=1}^{n}\left(X_{i}-\mu_{1}^{\prime}ight)^{k-j}ight]ight\} \text {. }\]

Prove that when \(j=3\) the term in the sum is \(O\left(n^{-2}ight)\) as \(n ightarrow \infty\).