Tropp, Joel A. (2011) Freedman’s Inequality for Matrix Martingales. Electronic Communications in Probability, 16 . pp. 262-270. ISSN 1083-589X. https://resolver.caltech.edu/CaltechAUTHORS:20110606-111746280
![]()
|
PDF
- Published Version
See Usage Policy. 164Kb |
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20110606-111746280
Abstract
Freedman’s inequality is a martingale counterpart to Bernstein’s inequality. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently established a natural extension of Freedman’s inequality that provides tail bounds for the maximum singular value of a matrix-valued martingale. This note describes a different proof of the matrix Freedman inequality that depends on a deep theorem of Lieb from matrix analysis. This argument delivers sharp constants in the matrix Freedman inequality, and it also yields tail bounds for other types of matrix martingales. The new techniques are adapted from recent work [Tro10b] by the present author.
Item Type: | Article | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Related URLs: |
| |||||||||
ORCID: |
| |||||||||
Additional Information: | © 2011 Institute of Mathematical Statistics. Submitted 15 January 2011, accepted in final form 25 March 2011. Published on: May 23, 2011. Research supported by ONR Award N00014-08-1-0883, DARPA Award N66001-08-1-2065, and AFOSR Award FA9550-09-1-0643. Roberto Oliveira introduced me to Freedman’s inequality and encouraged me to apply the methods from [Tro10b] to study the matrix extension of Freedman’s result. I would also like to thank Yao- Liang Yu, who pointed out an inconsistency in the proof of Theorem 2.3 and who proposed the argument in Lemma 3.2. Richard Chen and Alex Gittens have helped me root out (numerous) typographic errors. | |||||||||
Funders: |
| |||||||||
Subject Keywords: | Discrete-time martingale; large deviation; probability inequality; random matrix | |||||||||
Classification Code: | AMS 2000 Subject classification: Primary: 60B20. Secondary: 60F10, 60G42 | |||||||||
Record Number: | CaltechAUTHORS:20110606-111746280 | |||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20110606-111746280 | |||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
ID Code: | 23915 | |||||||||
Collection: | CaltechAUTHORS | |||||||||
Deposited By: | Tony Diaz | |||||||||
Deposited On: | 15 Jun 2011 18:20 | |||||||||
Last Modified: | 03 Oct 2019 02:51 |
Repository Staff Only: item control page