Let \(\left\{B_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(B_{n}\) has a \(\operatorname{Bernoulli}(\theta)\) distribution for all \(n \in \mathbb{N}\). Define a sequence of random variables \(\left\{X_{n}ight\}_{n=1}^{\infty}\) where

\[X_{n}=n^{-1} \sum_{k=1}^{n} B_{k}\]

which is the proportion of the first \(n\) BERNOULLI random variables that equal one. Prove that \(n^{1 / 2}\left(X_{n}-\thetaight) \xrightarrow{d} Z\) as \(n ightarrow \infty\) where \(Z\) has a \(\mathrm{N}\left[0, n^{-1} \theta(1-\theta)ight]\) distribution and that \(X_{n} \xrightarrow{p} \theta\) as \(n ightarrow \infty\). Using these conclusions, find the asymptotic distribution of \(n^{1 / 2}\left[X_{n}\left(1-X_{n}ight)-\theta(1-\theta)ight]\).