Electronic Communications in Probability

Freedman's inequality for matrix martingales

Joel Tropp

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Discrete-time martingale large deviation probability inequality random matrix

This work is licensed under aCreative Commons Attribution 3.0 License.


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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