= Solution
For the simple null, the restricted maximum-likelihood estimator is $\widetilde\theta=\theta_0$, and hence
$$
T_n=(n^{-1/2}S_n(\theta_0))^T
I_1(\theta_0)^{-1}
(n^{-1/2}S_n(\theta_0)).
$$
The individual scores are independent, have mean zero by part (ii), and have covariance $I_1(\theta_0)$. The multivariate central limit theorem gives
$$
n^{-1/2}S_n(\theta_0)
\xrightarrow{d}N_p(0,I_1(\theta_0)).
$$
After multiplying by $I_1(\theta_0)^{-1/2}$, the limit is standard normal in $\mathbb R^p$. The continuous mapping theorem therefore yields
$$
\boxed{T_n\xrightarrow{d}\chi_p^2.}
$$
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