# Positive Semidefinite Matrix Supermartingales

arXiv: https://arxiv.org/abs/2401.15567 · HTML: https://mlsanity.com/paper/2401.15567/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 28 Jan 2024.

## Builds on (most cited first, 13 of 13)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Confidence sequences for sampling without replacement](https://mlsanity.com/paper/2006.04347/) (2020)
- [Martingale Methods for Sequential Estimation of Convex Functionals and Divergences](https://mlsanity.com/paper/2103.09267/) (2021)
- [Catoni-style confidence sequences for heavy-tailed mean estimation](https://mlsanity.com/paper/2202.01250/) (2022)
- [User-Friendly Covariance Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/1811.01520/) (2018)
- [Randomized and Exchangeable Improvements of Markov's, Chebyshev's and Chernoff's Inequalities](https://mlsanity.com/paper/2304.02611/) (2023)
- [A composite generalization of Ville's martingale theorem using e-processes](https://mlsanity.com/paper/2203.04485/) (2022)
- [Huber-Robust Confidence Sequences](https://mlsanity.com/paper/2301.09573/) (2023)

## Abstract (arXiv)

We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative supermartingales and backward nonnegative submartingales, whose convergence and maximal inequalities are the theoretical foundations for a wide and ever-growing body of results in statistics, econometrics, and theoretical computer science. Our results lead to new concentration inequalities for either martingale-dependent or exchangeable random symmetric matrices under a variety of tail conditions, encompassing now-standard Chernoff bounds to self-normalized heavy-tailed settings. Further, these inequalities are usually expressed in the Loewner order, are sometimes valid simultaneously for all sample sizes or at an arbitrary data-dependent stopping time, and can often be tightened via an external randomization factor.