Hoeffding's inequalities for geometrically ergodic Markov chains on general state space

TL;DR

This paper derives Hoeffding's inequalities for geometrically ergodic Markov chains on general state spaces, providing bounds on large deviations of averages based on spectral properties.

Contribution

It introduces Hoeffding-type bounds for Markov chains with spectral gaps in $L^2$, extending classical inequalities to more general ergodic processes.

Findings

Bounds depend only on stationary mean, spectral gap, and support of $f$

Applicable to Markov chains with spectral gap in $L^2$

Provides probabilistic deviation bounds for averages along trajectories

Abstract

We consider Markov chain with spectral gap in $L^2$ space. Assume that $f$ is a bounded function. Then the probabilities of large deviations of average along trajectory satisfy Hoeffding's-type inequalities. These bounds depend only on the stationary mean, spectral gap and the end-points of support of $f$.