Matrix Freedman Inequality for Sub‐Weibull Martingales
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Bibliographic record
Abstract
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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it