When is Hotelling's $T^2$ asymptotically $F$-distributed?

arXiv:2610.12903v1 Announce Type: new Abstract: For Gaussian observations, a deterministic multiple of Hotelling's $T^2$ statistic has an exact $F$ distribution. This paper investigates when this approximation remains valid for an independent and identically distributed (i.i.d.) data matrix $\mathbf{X}\in \mathbb{R}^{p\times n}$ with a fixed, possibly heavy-tailed entry law $\xi$. Under symmetry and $\limsup_{n\rightarrow \infty} p/nx)=0$ when $p/n\rightarrow \gamma\in (0,1)$, and obtain the $F$ approximation as a consequence. Centered standardized Pareto laws show that the exponent $5/2$ is sharp for this universal absolute-tail sufficient condition, although balanced-tail nonsymmetric laws can retain the approximation at the boundary, which reveals that the asymmetry of the tails matters for the $F$ approximation.