Effective-dimension bounds in the central limit theorem for centered Euclidean balls

Abstract

We establish a nonasymptotic Gaussian approximation bound over centered Euclidean balls for sums of independent, not necessarily identically distributed, random vectors in \(\mathbb R^p\). Let \(S_n\) denote the sum, let \(Σ=\operatorname{Var}(S_n)\), and let \(G\sim N(0,Σ)\). Define the effective rank and the normalized squared spectral mass by \[ r(Σ)=\frac{\operatorname{tr}Σ}{\|Σ\|}, \qquad v(Σ)=\frac{\operatorname{tr}(Σ^2)}{\|Σ\|^2}. \] Under a uniform directional fourth-moment condition, covariance-balance assumptions, and the spectral conditions \(r(Σ)\le C_Σv(Σ)\) and \(v(Σ)\ge6\), we prove \[ \sup_{z\ge0} \left| \mathbb P\{\|S_n\|^2\le z\} - \mathbb P\{\|G\|^2\le z\} \right| \le C\,\frac{r(Σ)^{3/2}}{n}. \] The constant depends only on the fixed parameters in the assumptions and is independent of the ambient dimension and individual small eigenvalues of the covariance matrix. The bound is uniform at the endpoint \(v(Σ)=6\), without a logarithmic loss. Its main feature is the combination of the \(n^{-1}\) rate with dependence on effective rank, including this endpoint. Consequently, the bound can vanish when \(r(Σ)=o(n^{2/3})\), even when \(p\) is much larger than \(n\). The same estimate extends to random elements of a separable Hilbert space with trace class covariance. The proof builds on the Fourier and symmetrization methods of \citet{GoetzeZaitsev2014}, combined with moment matching, truncation, and comparison estimates that retain the covariance structure.