Concentration Inequalities for Sums of Markov Dependent Random Matrices

Abstract: We give Hoeffding and Bernstein-type concentration inequalities for the largest eigenvalue of sums of random matrices arising from a Markov chain. We consider time-dependent matrix-valued functions on a general state space, generalizing previous results that had only considered Hoeffding-type inequalities, and only for time-independent functions on a finite state space. In particular, we study a kind of noncommutative moment generating function, provide tight bounds on this object, and use a method of Garg et al. to turn this into tail bounds. Our proof proceeds spectrally, bounding the norm of a certain perturbed operator. In the process we make an interesting connection to dynamical systems and Banach space theory to prove a crucial result on the limiting behavior of our moment generating function that may be of independent interest.