  We prove a version of McDiarmid's bounded differences inequality for Markov
chains, with constants proportional to the mixing time of the chain. We also
show variance bounds and Bernstein-type inequalities for empirical averages of
Markov chains. In the case of non-reversible chains, we introduce a new
quantity called the "pseudo spectral gap", and show that it plays a similar
role for non-reversible chains as the spectral gap plays for reversible chains.
Our techniques for proving these results are based on a coupling construction
of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo
spectral gap generalises the multiplicative reversiblication approach of Jim
Fill.
