Suppose that \(\left\{X_{n}ight\}_{n=1}^{\infty}\) is a sequence of independent random variables from a common distribution that has mean \(\mu\) and variance \(\sigma^{2}\), such that \(E\left(\left|X_{n}ight|^{4}ight)<\infty\). Prove that

\[\begin{aligned}E\left(\bar{X}_{n}^{4}ight)= & n^{-4}\left[n \lambda+4 n(n-1) \mu \gamma+6 n(n-1)(n-2) \mu^{2}\left(\mu^{2}+\sigma^{2}ight)ight. \\& \left.+n(n-1)(n-2)(n-3) \mu^{4}ight] \\= & B(n)+n^{-3}(n-1)(n-2)(n-3) \mu^{4},\end{aligned}\]

where \(B(n)=O\left(n^{-1}ight)\) as \(n ightarrow \infty, \gamma=E\left(X_{n}^{3}ight)\), and \(\lambda=E\left(X_{n}^{4}ight)\).