Noncommutative martingale concentration inequalities

Abstract

We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving $L_p$-norm of the sum of a martingale difference.

18 pages, to appear in Illinois J. Math

References in corpus (2)

• Noncommutative Bennett and Rosenthal inequalities
• The Golden-Thompson inequality --- historical aspects and random matrix applications

Cited by in corpus (4)

• Etemadi and Kolmogorov inequalities in noncommutative probability spaces
• Noncommutative Blackwell-Ross martingale inequality
• A variance bound for a general function of independent noncommutative random variables
• Non-commutative Stein inequality and its applications