A geometric proof of multivariate self-normalized moderate deviations via ridge regularization
Abstract
Let X,X1,X2,… be i.i.d.\ random vectors in ℝd, where d≥1 is fixed. Assume that 𝔼 X=0, 𝔼 XX⊤=Id and 𝔼|X|3<∞. We prove a Cramér type moderate deviation theorem for a multivariate self-normalized radius statistic, that is, for every an=o(n1/6), sup0≤ x≤ an |(ℙ(Un,+≥ x))/(ℙ(χd≥ x))−1|→0, where χd is the Euclidean norm of a standard normal vector in ℝd and Un,+2=Sn⊤ An+Sn, Sn=∑j=1nXj, An=∑j=1nXjXj⊤. Here An+ denotes the Moore--Penrose inverse. We further prove that the same result holds for the corresponding Hotelling T2 statistic. This work closes the moment gap between the previously known finite (3+δ) moment condition for Hotelling moderate deviations and the finite third moment boundary. Our proof adopts a geometric approach. Truncation at the finite-third-moment scale yields a truncated covariance matrix whose concentration error is negligible relative to the target χd tail. Uniform projected self-normalized moderate deviations are then averaged over directions and, after a whitening map with a ridge term, become a spherical cap transform of the radial distribution. A local Abel inversion recovers the radial tail, and deterministic comparison removes the ridge and the truncation.
Related
- Normal approximation and self-normalized moderate deviations for the minimal random walk with random step sizes
- Moderate deviations and laws of the iterated logarithm for geometric functionals in the sparse regime
- Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes
- Moderate deviations for the Maki--Thompson rumour model
- Normal approximation, moderate and large deviations for the elephant random walk with stops and random step sizes
BibTeX
@misc{jing2026geometric,
title = {{A geometric proof of multivariate self-normalized moderate deviations via ridge regularization}},
author = {Bing-Yi Jing and Yiming Liu and Shaochen Wang and Wang Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05822},
eprint = {2611.05822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05822},
}
RIS
TY - UNPB
TI - A geometric proof of multivariate self-normalized moderate deviations via ridge regularization
AU - Bing-Yi Jing
AU - Yiming Liu
AU - Shaochen Wang
AU - Wang Zhou
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05822
UR - https://arxiv.org/abs/2611.05822
AB - Let X,X_1,X_2,… be i.i.d.\ random vectors in ℝ^d, where d≥1 is fixed. Assume that 𝔼 X=0, 𝔼 XX^⊤=I_d and 𝔼|X|^3<∞. We prove a Cramér type moderate deviation theorem for a multivariate self-normalized radius statistic, that is, for every a_n=o(n^1/6), sup_0≤ x≤ a_n |(ℙ(U_n,+≥ x))/(ℙ(χ_d≥ x))−1|→0, where χ_d is the Euclidean norm of a standard normal vector in ℝ^d and U_n,+^2=S_n^⊤ A_n^+S_n, S_n=∑_j=1^nX_j, A_n=∑_j=1^nX_jX_j^⊤. Here A_n^+ denotes the Moore--Penrose inverse. We further prove that the same result holds for the corresponding Hotelling T^2 statistic. This work closes the moment gap between the previously known finite (3+δ) moment condition for Hotelling moderate deviations and the finite third moment boundary. Our proof adopts a geometric approach. Truncation at the finite-third-moment scale yields a truncated covariance matrix whose concentration error is negligible relative to the target χ_d tail. Uniform projected self-normalized moderate deviations are then averaged over directions and, after a whitening map with a ridge term, become a spherical cap transform of the radial distribution. A local Abel inversion recovers the radial tail, and deterministic comparison removes the ridge and the truncation.
ER -