arXiv:2611.08074v1 Announce Type: new Abstract: Let $\mathbf X_1,\ldots,\mathbf X_n$ be independent random vectors in $\mathbb R^d$ with common mean $\boldsymbol\mu_n$, where $d\ge2$ is a fixed integer. Let $\bar{\mathbf X}_n=n^{-1}\sum_{i=1}^n\mathbf X_i$ denote the sample mean and let \[ \widehat\Sigma_n=\frac1n\sum_{i=1}^n (\mathbf X_i-\bar{\mathbf X}_n) (\mathbf X_i-\bar{\mathbf X}_n)^\top \] denote the sample covariance matrix. Assume that the average covariance matrix is positive definite and that $\widehat\Sigma_n$ is nonsingular almost surely. For testing $H_0:\boldsymbol\mu_n=\boldsymbol\mu_0$, the Hotelling statistic is \[ T^2_{H,n} =n(\bar{\mathbf X}_n-\boldsymbol\mu_0)^\top \widehat\Sigma_n^{-1} (\bar{\mathbf X}_n-\boldsymbol\mu_0). \] Under $H_0$, we establish finite-sample Cram\'er-type moderate deviation results and uniform Berry--Esseen bounds under finite $p$th moments for $2Source：https://arxiv.org/abs/2611.08074
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