Exponential inequalities for nonstationary Markov Chains
This work addresses a gap in machine learning theory for non-independent, nonstationary data, enabling learning techniques for applications like periodic time series, though it is incremental as it builds on prior stationary results.
The authors tackled the lack of exponential inequalities for nonstationary Markov chains, extending existing tools to derive a Bernstein-type inequality and risk bounds for predicting periodic autoregressive processes with unknown periods.
Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behavior is non-stationary. In this paper, we extend the basic tools of Dedecker and Fan (2015) to nonstationary Markov chains. As an application, we provide a Bernstein-type inequality, and we deduce risk bounds for the prediction of periodic autoregressive processes with an unknown period.