Freedman’s Inequality for Matrix Martingales

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. 1 An Introduction to Freedman’s Inequality The Freedman inequality [Fre75, Thm. (1.6)] is a martingale extension of the Bernstein inequality. This result demonstrates that a martingale exhibits normal-type concentration near its mean value on a scale determined by the predictable quadratic variation, and the upper tail has Poisson-type decay on a scale determined by a uniform bound on the difference sequence. Oliveira [Oli10, Thm. 1.2] proves that Freedman’s inequality extends, in a certain form, to the matrix setting. The purpose of this note is to demonstrate that the methods from the author’s paper [Tro10b] can be used to establish a sharper version of the matrix Freedman inequality. Furthermore, this approach offers a transparent way to obtain other probability inequalities for adapted sequences. Let us introduce some notation and background on martingales so that we can state Freedman’s original result rigorously. Afterward, we continue with a statement of our main results and a presentation of the methods that we need to prove the matrix generalization. 1RESEARCH SUPPORTED BY ONR AWARD N00014-08-1-0883, DARPA AWARD N66001-08-1-2065, AND AFOSR AWARD FA9550-09-1-0643.