Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables following a \(\mathrm{N}(\theta, 1)\) distribution.

a. Prove that if \(\theta eq 0\) then \(\delta\left\{\left|\bar{X}_{n}ight|:\left[0, n^{-1 / 4}ight)ight\} \xrightarrow{p} 0\) as \(n ightarrow \infty\).

b. Prove that if \(\theta=0\) then \(\delta\left\{\left|\bar{X}_{n}ight|:\left[0, n^{-1 / 4}ight)ight\} \xrightarrow{p} 1\) as \(n ightarrow \infty\).