A geometric proof of multivariate self-normalized moderate deviations via ridge regularization

Take i.i.d. vectors \(X_j\) in \(\mathbb{R}^d\) with mean zero, identity covariance and \(\mathbb{E}\|X\|^3 < \infty\). With \(S_n = \sum_j X_j\) and \(A_n = \sum_j X_j X_j^\top\), the statistic is \(U_{n,+}^2 = S_n^\top A_n^+ S_n\), where \(A_n^+\) is the Moore–Penrose inverse. The theorem states that for every \(a_n = o(n^{1/6})\), \[\sup_{0 \le x \le a_n} \left| \frac{\mathbb{P}(U_{n,+} \ge x)}{\mathbb{P}(\chi_d \ge x)} - 1 \right| \to 0.\]

The same result holds for the Hotelling \(T^2\) statistic, which closes the gap between the known \((3+\delta)\)-moment condition and the third-moment boundary. The proof is geometric. It truncates at the third-moment scale, averages uniform projected moderate deviations over directions, turns them into a spherical cap transform of the radial law after a whitening map with a ridge term, and recovers the radial tail by a local Abel inversion.