A tail inequality for suprema of unbounded empirical processes with applications to Markov chains

TL;DR

This paper introduces a tail inequality for the supremum of unbounded empirical processes with applications to Markov chains, providing new bounds and inequalities for empirical processes and symmetric statistics in this context.

Contribution

It develops a tail inequality for unbounded empirical processes with finite _ norms and applies it to derive bounds for Markov chains and symmetric statistics.

Findings

Derived a tail inequality for empirical processes with _ norms.

Applied the inequality to geometrically ergodic Markov chains.

Established a bounded difference inequality for symmetric statistics of Markov chains.

Abstract

We present a tail inequality for suprema of empirical processes generated by variables with finite $\psi_\alpha$ norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.