Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a distribution \(F\) with \(E\left(\left|X_{1}ight|^{2 k}ight)<\infty\).

a. Prove that

\[\begin{array}{r}E\left(\hat{\mu}_{k}^{2}ight)=\mu_{k}^{2}+n^{-1}\left[\mu_{2 k}-\mu_{k}^{2}-2 k\left(\mu_{k}^{2}+\mu_{k-1} \mu_{k+1}ight)+k^{2} \mu_{k-1}^{2} \mu_{2}+ight. \\\left.k(k-1) \mu_{k} \mu_{k-2} \mu_{2}ight]+O\left(n^{-2}ight),\end{array}\] as \(n ightarrow \infty\).

b. Use this result to prove that \[V\left(\hat{\mu}_{k}ight)=n^{-1}\left(\mu_{2 k}-\mu_{k}^{2}-2 k \mu_{k-1} \mu_{k+1}+k^{2} \mu_{2} \mu_{k-1}^{2}ight)+O\left(n^{-2}ight),\]
as \(n ightarrow \infty\).