Electronic Communications in Probability

Freedman's inequality for matrix martingales

Joel Tropp

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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 by the present author.

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 25, 262-270.

Dates
Accepted: 23 May 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261981

Digital Object Identifier
doi:10.1214/ECP.v16-1624

Mathematical Reviews number (MathSciNet)
MR2802042

Zentralblatt MATH identifier
1225.60017

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60F10: Large deviations 60G42: Martingales with discrete parameter

Keywords
Discrete-time martingale large deviation probability inequality random matrix

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Tropp, Joel. Freedman's inequality for matrix martingales. Electron. Commun. Probab. 16 (2011), paper no. 25, 262--270. doi:10.1214/ECP.v16-1624. https://projecteuclid.org/euclid.ecp/1465261981


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