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Concentration inequalities for random dynamical systems

Graccyela Salcedo

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.

Concentration inequalities for random dynamical systems

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.

Paper Structure

This paper contains 25 sections, 30 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Random dynamical systems
  3. Notations
  4. Main Result
  5. Concentration inequalities
  6. Auxiliary results
  7. Proofs of the main concentration results
  8. Weakly contracting on average RDSs
  9. Key properties
  10. Almost-sure central limit theorem
  11. Step 1: Convergence in mean.
  12. Step 2: Almost-sure convergence.
  13. Special case: contraction on average
  14. Applications of concentration inequalities
  15. Synchronization in time averages
  16. ...and 10 more sections

Key Result

Theorem A

Fix $n \in \mathbb{N}$. Let $\gamma_0, \gamma_1, \dots, \gamma_n$ be nonnegative real numbers, with at least one strictly positive, and set Then, for every function $\varphi \in \mathrm{Lip}_{d+\varrho}\left((\mathcal{G} \times M)^{n+1}, \gamma\vert_0^n \right)$, for all $x \in M$ and all $t > 0$, we have

Theorems & Definitions (63)

  • Theorem A
  • Example 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • ...and 53 more