Concentration and moment inequalities for sums of independent heavy-tailed random matrices
Moritz Jirak, Stanislav Minsker, Yiqiu Shen, Martin Wahl
TL;DR
This work extends Fuk-Nagaev and Rosenthal-type inequalities to sums of independent random matrices with heavy-tailed norms, emphasizing intrinsic dimensionality through the effective rank rather than ambient dimension. By combining truncation, matrix Bernstein-type bounds, and Banach-space tools, the authors derive sharp tail and moment bounds that adapt to heavy tails and finite low-order moments. They then apply these results to matrix subsampling, covariance estimation, and empirical eigenvector estimation, obtaining nonasymptotic bounds that improve over prior results in terms of dependence on effective rank and relative ranks. The findings have direct implications for high-dimensional covariance analysis and spectral inference under heavy-tailed data, with concrete bounds for submatrix norms, covariance error, and eigenvector perturbations. Overall, the paper provides a unified, dimension-aware probabilistic toolkit for heavy-tailed random matrices with broad applications in statistics and signal processing.
Abstract
We prove Fuk-Nagaev and Rosenthal-type inequalities for sums of independent random matrices, focusing on the situation when the norms of the matrices possess finite moments of only low orders. Our bounds depend on the ``intrinsic'' dimensional characteristics such as the effective rank, as opposed to the dimension of the ambient space. We illustrate the advantages of such results through several applications, including new moment inequalities for sample covariance matrices and their eigenvectors when the underlying distribution is heavy-tailed. Moreover, we demonstrate that our techniques yield sharpened versions of moment inequalities for empirical processes.