A Hoeffding inequality for Markov chains

We prove deviation bounds for the random variable \(\sum_{i=1}^{n} f_i(Y_i)\) in which \(\{Y_i\}_{i=1}^{\infty}\) is a Markov chain with stationary distribution and state space \([N]\), and \(f_i: [N] \rightarrow [-a_i, a_i]\). Our bound improves upon previously known bounds in that the dependence is on \(\sqrt{a_1^2+\cdots+a_n^2}\) rather than \(\max_{i}\{a_i\}\sqrt{n}.\) 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 \(\infty\)-norm of a random matrix whose entries are obtained from a Markov chain.