Exponential inequalities for nonstationary Markov Chains

TL;DR

This paper extends exponential inequalities to nonstationary Markov chains, enabling better analysis of non-stationary time series like periodic autoregressive processes in machine learning.

Contribution

It generalizes existing tools to nonstationary Markov chains, providing new Bernstein-type inequalities and risk bounds for periodic time series prediction.

Findings

Derived Bernstein-type inequality for nonstationary Markov chains

Provided risk bounds for predicting periodic autoregressive processes

Extended exponential inequalities to nonstationary, non-i.i.d. data

Abstract

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.