Matrix Freedman Inequality for Sub‐Weibull Martingales

ABSTRACT In this paper, we establish a matrix Freedman inequality for martingales with sub‐Weibull tails. Under conditional control of the increments, the top eigenvalue admits a non‐asymptotic tail bound with explicit, dimension‐aware constants. Via Hermitian dilation, our result extends to rectangular matrices, recovers the sub‐Gaussian case at and admits a time‐uniform (supremum‐over‐time) form. Relative to recent Bennett/Bernstein bounds for sub‐Weibull matrix martingales, our thresholds depend only on a variance proxy and a radius. Concretely, in high‐confidence regimes with , these new thresholds match or improve the corresponding modern envelopes at the same confidence level. We illustrate the utility of our bound in two applications: (i) self‐normalized confidence sets for stochastic linear bandits with heavy‐tailed noise and (ii) operator‐norm error bounds for covariance estimation. We corroborate the theory and highlight constant‐level effects through simulations over a range of tail indices and variance levels.