I am dealing with an $n$-dimensional random variable $\hat{P}$ for which I know that $$\sqrt{n}(\hat{P}-P) \to^d \mathcal{N}(\mathbf{0},\Sigma).$$ I could also estimate the covariance matrix, $\Sigma$. I am wondering if this information could help me define regions of confidence for the "mean" $P$, as one would do using Hotelling's T-squared statistic had the vector been the sum of independent and identically distributed random variables and $\Sigma$ was estimated the usual way ? More precisely, can one use Hotelling's test (https://en.wikipedia.org/wiki/Hotelling%27s_T-squared_distribution) if only asymptotic convergence in distribution towards a Gaussian distribution is known to be true, with $\hat{P}$ not necesarrily being a sum of random variables ?
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