A Hoeffding inequality for Markov chains

We prove deviation bounds for the random variable Sigma(n)(i=1) f(i)(Y-i) in which {Y-i}(i=1)(infinity )is a Markov chain with stationary distribution and state space [N], and f(i) : [N] -> [-a(i), a(i)]. Our bound improves upon previously known bounds in that the dependence is on root a(1)(2) + ... + a(n)(2) rather than max(i) {a(i)}root n. We also prove deviation bounds for certain types of sums of vector-valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten infinity-norm of a random matrix whose entries are obtained from a Markov chain.