Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a distribution \(F\).

a. Prove that if \(E\left(\left|X_{1}ight|^{k}ight)<\infty\) then \(\hat{\mu}_{k}^{\prime}\) is an unbiased estimator of \(\mu_{k}^{\prime}\).

b. Prove that if \(E\left(\left|X_{1}ight|^{2 k}ight)<\infty\) then the standard error of \(\hat{\mu}_{k}^{\prime}\) is \(n^{-1 / 2}\left(\mu_{2 k}^{\prime}-ight.\) \(\left.\mu_{k}^{\prime}ight)^{1 / 2}\).

c. Prove that if \(E\left(\left|X_{1}ight|^{k}ight)<\infty\) then \(\hat{\mu}_{k}^{\prime} \xrightarrow{\text { a.c. }} \mu_{k}^{\prime}\) as \(n ightarrow \infty\).