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              <p>This paper concerns error bounds for recursive equations subject to Markovian disturbances.Motivating  examples  abound  within  the  fields  of  Markov  chain  Monte  Carlo  (MCMC)  andReinforcement Learning (RL), and many of these algorithms can be interpreted as special casesof  stochastic  approximation  (SA).  It  is  argued  that  it  is  not  possible  in  general  to  obtain  aHoeffding bound on the error sequence,  even when the underlying Markov chain is reversibleand geometrically ergodic, such as the M/M/1 queue.  This is motivation for the focus on meansquare error bounds for parameter estimates.  It is shown that mean square error achieves theoptimal rate ofO(1/n), subject to conditions on the step-size sequence.  Moreover, the exactconstants in the rate are obtained, which is of great value in algorithm design.</p>
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