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Let $X_1,\ldots,X_n$ be an i.i.d sample from a distribution on $\mathbb R^p$ with mean $\mu = 0 \in \mathbb R^p$ and $p$-by-$p$ covariance matrix $\Sigma$ of rank $r \le p$. Consider the centered Hotelling $T^2$ statistic defined by $T_n^2 := n\hat{\mu}_n^T\Sigma_n^{\dagger}\hat{\mu}_n$, $\hat{\mu}_n$ and $\hat{\Sigma}_n$ are the empirical mean vector and covariance matrix, respectively. It is well-known that $H_n$ converges in law to $\chi^2_{(r)}$, a chi-squared distribution with $r$ degrees of freedom (Sepanski 1994).

Question

What are good non-asymptotic bounds for $P(T_n^2 \ge \epsilon)$ ?

I'm hoping for a Berry-Esseen-type bound of the form $\sup_{\epsilon > 0}|P(T^2_n \ge \epsilon) - P(\chi^2_{(r)} \ge \epsilon)| \le C/n^\delta$, for some absolute constants $C > 0$ and $\delta > 0$ (rough guess: $\delta \in [1/6,1/2]$).

Observation

A non-asymptotic tail-bound is available in the noncentral case $\mu \ne 0$. See Hotelling non-centralisé, e.g Theorem 4.24 of this paper.

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