Concentration inequalities for random dynamical systems

TL;DR

This paper derives concentration inequalities for random dynamical systems with Lipschitz observables, providing deviation bounds for various quantities under weak contraction conditions, applicable to systems like circle diffeomorphisms and linear cocycles.

Contribution

It introduces new concentration inequalities for RDSs under weak average contraction, covering quantities like synchronization, empirical measures, and Lyapunov exponents.

Findings

Deviation bounds for time-average synchronization

Concentration inequalities for empirical measures and Birkhoff sums

Finite-time Lyapunov exponents concentration results

Abstract

We establish concentration inequalities for random dynamical systems (RDSs), assuming that the observables of interest are separately Lipschitz. Under a weak average contraction condition, we obtain deviation bounds for several random quantities, including time-average synchronization, empirical measures, Birkhoff sums, and correlation dimension estimators. We present concrete classes of RDSs to which our main results apply, such as finitely supported diffeomorphisms on the circle and projective systems induced by linear cocycles. In both cases, we obtain concentration inequalities for finite-time Lyapunov exponents.