 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[No.] Part (c) shows that $n(\widehat\theta-\theta_0)=O_{\mathbb P}(1)$, and therefore
$$
\sqrt n(\widehat\theta-\theta_0)
=\frac{n(\widehat\theta-\theta_0)}{\sqrt n}
\xrightarrow{\mathbb P}0.
$$
Every distributional <limit> is the point mass at zero and has variance zero. The regular maximum-likelihood central <limit> theorem fails because the support depends on $\theta$.

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