Bernstein-type and Bennett-type inequalities for unbounded matrix martingales
Alexey Kroshnin, Alexandra Suvorikova
TL;DR
This work develops Bernstein-type and Bennett-type concentration bounds for matrix-valued martingales with unbounded observations in $\mathbb{H}(d)$ by bounding the conditioned $ψ_\alpha$-Orlicz norms of the increments. It introduces a conditioned Orlicz-norm framework and derives explicit ambient-dimension bounds, as well as dimension-free bounds that depend on the effective rank $r(\boldsymbol{\Sigma})$ rather than the ambient dimension. The paper further provides practical corollaries, including an empirical Bernstein inequality and a matrix McDiarmid inequality, highlighting data-driven variance estimation and robustness to dependence. The results unify and extend existing matrix concentration theory to the unbounded setting and offer regime-specific tail behavior (sub-Gaussian, sub-Poisson, sub-exponential) with explicit constants, enabling sharper analysis in high-dimensional stochastic processes.
Abstract
We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued martingale processes with unbounded observations from the Hermitian space $\mathbb{H}(d)$. Specifically, we assume that the $ψ_α$-Orlicz (quasi-)norms of their difference process are bounded for some $α> 0$. Further, we generalize the obtained result by replacing the ambient dimension $d$ with the effective rank of the covariance of the observations. To illustrate the applicability of the results, we prove several corollaries, including an empirical version of Bernstein's inequality and an extension of the bounded difference inequality, also known as McDiarmid's inequality.