The Berry-Esseen Bound for High-dimensional Self-normalized Sums (2501.08979v1)

Abstract: This manuscript studies the Gaussian approximation of the coordinate-wise maximum of self-normalized statistics in high-dimensional settings. We derive an explicit Berry-Esseen bound under weak assumptions on the absolute moments. When the third absolute moment is finite, our bound scales as $\log^{5/4}(d)/n^{1/8}$ where $n$ is the sample size and $d$ is the dimension. Hence, our bound tends to zero as long as $\log(d)=o(n^{1/10})$. Our results on self-normalized statistics represent substantial advancements, as such a bound has not been previously available in the high-dimensional central limit theorem (CLT) literature.

• The paper establishes a Berry-Esseen bound for high-dimensional self-normalized sums, scaling as log^(5/4)(d)/n^(1/8) and allowing finite third absolute moments, improving prior limits.
• The research introduces Best and Moment Matching Gaussian approximation techniques, showing the convergence of the bound y can be achieved even without a finite second moment.
• This work significantly impacts high-dimensional probability theory and practical inference, especially for bootstrap procedures and quantitative risk analysis in fields like genomics, machine learning, and finance.

Insights on the Berry-Esseen Bound for High-dimensional Self-normalized Sums

This research paper explores the intricacies of the Gaussian approximation of coordinate-wise maxima in the context of high-dimensional self-normalized statistics. The authors achieve significant strides in the theoretical underpinning of the high-dimensional Central Limit Theorem (CLT) by deriving an explicit Berry-Esseen bound. The advancement represents a pioneering effort to address the gap left by previous research within the high-dimensional probabilistic and statistical framework.

Summary of Main Contributions

The core contribution of this research is the establishment of a Berry-Esseen bound that applies to self-normalized sums in high-dimensional settings. The analysis is structured around a less restrictive scenario concerning the absolute finite moments, specifically accommodating scenarios where the third absolute moment is finite. The derived bound scales as $\log^{5/4}(d)/n^{1/8}$, where $n$ refers to the sample size and $d$ indicates the dimension. Consequently, the bound tends to zero provided that $\log(d) = o(n^{1/10})$, marking an improvement over prior limits within the literature that did not accommodate such dimensionality growth relative to sample size.

The authors introduce two Gaussian approximation techniques to accommodate different levels of moment assumptions:

1. Best Gaussian Approximation is defined to explore the infimum over Gaussian distributions with correlation structures.
2. Moment Matching Gaussian Approximation leverages the covariance structure inherent to the distribution of $X_1$ up to the second moment.

The research reveals that the newly introduced $\Delta_n$ can converge to zero sans a finite second moment, expanding its applicability compared to previous approaches.

Theoretical and Practical Implications

The implications of this research are manifold. Theoretically, it enriches the landscape of high-dimensional probability theory, enabling a better understanding of the nuances involved in self-normalized statistics beyond conventional array settings. Practically, the findings bear relevance for inferential methodologies, especially within the context of bootstrap procedures that rely on Gaussian approximations. The boundary conditions extracted from the Berry-Esseen bound can aid in implementing robust quantitative risk analyses in fields requiring inference in high dimensions, such as genomics, machine learning, and finance.

Speculations on Future Directions

The framework set forth by this paper paves the way for future explorations into the efficiency and refinement of statistical bounds in expanding dimensional settings. Specifically, reducing the dependency on $\log^5(ed)$ to more refined approximations, akin to those in settings without self-normalization, remains an open avenue. Additionally, incorporating advanced statistical techniques to cater to dependencies across indices could optimize existing results.

Furthermore, consistent with the growing emphasis on non-Euclidean data structures and models in artificial intelligence, extending the principles of self-normalization to more complex geometric inequalities represents a nascent area for future research.

In conclusion, this paper fundamentally enriches the high-dimensional statistical paradigm by setting new benchmarks in approximation precision and offering novel avenues for expansive CLT applications. Its role in refining theoretical models and crafting robust statistical techniques is both timely and indispensable, given the relentless push towards higher-dimensional data analytics.