Bernstein-type and Bennett-type inequalities for unbounded matrix martingales

Abstract

We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued supermartingale processes with unbounded observations. Specifically, we assume that the $psi_alpha$- Orlicz (quasi-)norms of their difference process are bounded for some $alpha > 0$. As corollaries, we prove an empirical version of Bernstein's inequality and an extension of the bounded differences inequality, also known as McDiarmid's inequality.