= Solution
Under the diagonal covariance restriction, the <profile log-likelihood> separates into $p$ terms,
$$
-\frac n2\sum_j\left(\log v_j+\frac{S_{jj}}{v_j}\right)+C.
$$
Differentiating each term gives the restricted <maximum-likelihood estimator> $\widehat V_0=D=\operatorname{diag}(S_{11},\ldots,S_{pp})$. Both fitted trace terms equal $p$, so the <likelihood-ratio test statistic> is
$$
\boxed{W=2\{\ell_p(S)-\ell_p(D)\}
=n\log\frac{\prod_jS_{jj}}{\det S}=-n\log\det R,}
$$
where $R=D^{-1/2}SD^{-1/2}$ is the <sample correlation matrix>. The statistic is nonnegative by the <Hadamard inequality>; large values indicate dependence. Equivalently, the likelihood ratio is $(\det S/\det D)^{n/2}$ and rejection is for small values of this ratio.
For fixed $p$ and a positive-definite true covariance, the <Wilks theorem> says that twice the maximized log-likelihood difference for nested regular models converges under the null to a <chi-squared distribution> with degrees of freedom equal to the dimension difference. The unrestricted covariance has $p(p+1)/2$ parameters and the restricted covariance has $p$; the unknown mean has the same $p$ parameters in both. Therefore the <Gaussian covariance diagonality likelihood-ratio test> of asymptotic level $\alpha$ is
$$
\boxed{\text{reject if }W>\chi^2_{p(p-1)/2,\,1-\alpha}.}
$$
Within the <multivariate normal distribution>, diagonal covariance also means independent coordinates.
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