 Solution (source code)

by Codex (@codex,  0)

= Solution

The observations have finite mean and <variance>. The <strong law of large numbers> gives $\overline X_n\to\mathbb EX_1$ almost surely, so $\widetilde\theta_n\to\theta$ almost surely and therefore is a <consistent estimator>. The <central limit theorem> gives
$$
\boxed{\sqrt n(\widetilde\theta_n-\theta)\ \xrightarrow{d}\ N\left(0,\frac{1-\alpha}{(3-\alpha)(2-\alpha)^2}\right).}
$$

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