USER-FRIENDLY TAIL BOUNDS FOR SUMS OF RANDOM MATRICES

Abstract

This work presents probability inequalities for sums of independent, random, self-adjoint
\nmatrices. The results frame simple, easily verifiable hypotheses on the summands, and they
\nyield strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum.
\nTail bounds for the norm of a sum of rectangular matrices follow as an immediate corollary, and
\nsimilar techniques yield information about matrix-valued martingales.
\nIn other words, this paper provides noncommutative generalizations of the classical bounds
\nassociated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The
\nmatrix inequalities promise the same ease of use, diversity of application, and strength of conclusion
\nthat have made the scalar inequalities so valuable.