Freedman's inequality for matrix martingales

Details Freedman's inequality for matrix martingales Freedman's inequality for matrix martingales

2011-01-16

arxiv.org/abs/1101.3039v1

Mathematics

Probability

8 pages. This note contains some martingale results that were presented in "User-friendly tail bounds for sums of random matri

Scientific paper

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.

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