Exponential inequalities for nonstationary Markov Chains

Exponential and concentration inequalities are corner stones of machine learning theory. The first distribution-free bounds on the Empirical Risk Minimiser (ERM), proven by Vapnik and Cervnonenkis in the early 70s, are based on Hoeffding's inequality, see Vapnik (1998). Model selection techniques rely heavily on concentration inequalities (Massart (2007)). We defer the reader to Boucheron et al. (2013) for an overview on concentration inequalities. However, all the results in these references are in the case of i.i.d random variables. Many extensions of Hoeffding and Bernstein's inequalities were proposed for dependent observations: see Catoni (2003); Bertail and Clémençon (2010); Joulin and Ollivier (2010); Dedecker and Fan (2015); Fan et al. (2018) under