# Non-asymptotic tail-bounds for Hotelling $T^2$ statistic

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.