Nonconventional Limit Theorems and Random Dynamics

The following sections are included:

• Preface
• Contents

The following sections are included:

• Introduction: local (strong) dependence structure
• Stein’s method for normal aproximation
  • A short introduction
  • Normal approximation for graphical indexation
  • Weak local dependence coefficients
  • Relations with more familiar mixing coefficients
• Nonconventional CLT with convergence rates
  • Assumptions and main results
  • Asymptotic variance
  • CLT with convergence rate
  • The associated strong dependency graphs
  • Expectation estimates
  • Proof of Theorem 1.3.7
  • Back to graphical indexation
• General Stein’s estimates: proofs
  • Proof of Theorem 1.2.1
  • Proof of Theorem 1.2.2
• Stein’s method for diffusion approximations
  • A functional CLT via Stein’s method
  • Finite dimensional convergence rate
• A nonconventional functional CLT
• Extensions to nonlinear indexes
  • Preliminaries
  • Another example with nonlinear indexes

The following sections are included:

• Introduction
• Local central limit theorem via Fourier analysis
• Nonconventional LLT for Markov chains by reduction to random dynamics
• Markov chains with densities
  • Basic assumptions and CLT
  • Characteristic functions estimates
• Markov chains related to dynamical systems
  • Locally distance expanding maps and transfer operators
  • Inverse branches, the pairing property and periodic points
  • Thermodynamic formalism constructions and the associated Markov chains
  • Relations between dynamical systems and Markov chains
  • Mixing and approximation assumptions
  • Asymptotic variance and the CLT
• Statement of the local limit theorem
• The associated random transfer operators
  • Random complex RPF theorem
  • Distortion properties
  • Reduction to random dynamics
• Decay of characteristic functions for small t’s
  • The random pressure function
  • The derivatives of the pressure
  • Pressure near 0
  • Norms estimates: employing the pressure
  • Remarks
• Decay of characteristic functions for large t’s
  • Basic estimates and strategy of the proof
  • Probabilities of large number of visits to open sets
  • Segments of periodic orbits
  • Parametric continuity of transfer operators
  • Quasi compactness of R_it, lattice and non-lattice cases
  • Norms estimates
• The periodic point approach to quenched and annealed dynamics
• Extensions to dynamical systems
  • Subshifts of finite type
  • The strategy of the proof
  • Local limit theorem: one sided case
  • Two sided case
  • Reduction to one sided case

We extend the nonconventional law of large numbers, the central limit theorem and Poisson limit theorems from [42], [45] and [43] to sums of the form ${\displaystyle {\sum }_{n=1}^{N}F\left({\xi }_{p_{1}n+q_{1}N},{\xi }_{p_{2}n+q_{2}N},\dots ,{\xi }_{p_{\mathcal{l}}n+q_{\mathcal{l}}N}\right)}$ where $\left\{{\xi }_n,n\ge 0\right\}$ is a sufficiently fast mixing stochastic process with some moment conditions and stationarity properties while F is a continuous function with polynomial growth and certain regularity properties. We discuss also the crucial question on positivity of the variance in this central limit theorem.

In this chapter we will prove a random complex Ruelle-Perron-Frobenius type theorem under rather general assumptions which will be verified in Chapter 5 for random locally expanding covering maps and in Chapter 6 for random complex integral operators.

The following sections are included:

• Random locally expanding covering maps
• Transfer operators
• Real and complex cones
• RPF triplets
• Properties of cones: proof of Theorem 5.3.1
• Properties of transfer operators: proof of Theorem 5.4.1 (i)
  • Continuity and analyticity
  • Analyticity in z
• Real Hilbert metric estimates
  • General estimates
  • Real cones invariances and diameter of image estimates
• Comparison of real and complex operators
• Complex image diameter estimates
• Real RPF triplets and Gibbs measures
• Complex Gibbs functionals
• The largest characteristic exponents
• Extension to the unbounded case for minimal systems

The following sections are included:

• Integral operators
• Real and complex cones
• The RPF theorem for integral operators
• Properties of cones: proof of Theorem 6.2.1
• Real cones: invariance and diameter of image estimates
• Comparison of real and complex operators
• Complex image diameter estimates

The following sections are included:

• The “conventional” case: preliminaries and main results
  • Self normalized Berry-Esseen theorem
  • Local limit theorem
• Pressure near 0
• A fiberwise (self normalized) Berry-Esseen theorem: proof
• A fiberwise local (central) limit theorem: proof
  • Characteristic functions estimates for small t’s
  • Characteristic functions estimates for large t’s: covering maps
  • Characteristic functions estimates for large t’s: Markov chains
  • An extension
• Nonconventional limit theorems for processes in random environment
  • Central limit theorem
  • Local limit theorem

In this appendix we describe the theory of complex projective Hilbert metrics associated with complex cones, and the contraction properties of linear maps between such cones with respect to the corresponding Hilbert metrics. We introduce the basic notations and tools which were developed first in [58] and then in [18]. Still, for readers’ convenience, we first recall the definition and basic properties of real cones and the real Hilbert metric associated with it (see [7], [49], [13], [18], [19] and [58]).

The following sections are included:

• Bibliography
• Index