Local Limit Theorems for Inhomogeneous Markov Chains (Lecture Notes in Mathematics)

Acknowledgments
Contents
Notation
1 Overview
	1.1 Setup and Aim
	1.2 The Obstructions to the Local Limit Theorems
	1.3 How to Show that the Obstructions Do Not Occur
	1.4 What Happens When the Obstructions Do Occur
		1.4.1 Lattice Case
		1.4.2 Center-Tight Case
		1.4.3 Reducible Case
	1.5 Some Final Words on the Setup of this Work
	1.6 Prerequisites
	1.7 Notes and References
2 Markov Arrays, Additive Functionals, and Uniform Ellipticity
	2.1 The Basic Setup
		2.1.1 Inhomogeneous Markov Chains
		2.1.2 Inhomogeneous Markov Arrays
		2.1.3 Additive Functionals
	2.2 Uniform Ellipticity
		2.2.1 The Definition
		2.2.2 Contraction Estimates and Exponential Mixing
		2.2.3 Bridge Probabilities
	2.3 Structure Constants
		2.3.1 Hexagons
		2.3.2 Balance and Structure Constants
		2.3.3 The Ladder Process
	2.4 γ-Step Ellipticity Conditions
	*2.5 Uniform Ellipticity and Strong Mixing Conditions
	2.6 Reduction to Point Mass Initial Distributions
	2.7 Notes and References
3 Variance Growth, Center-Tightness, and the CentralLimit Theorem
	3.1 Main Results
		3.1.1 Center-Tightness and Variance Growth
		3.1.2 The Central Limit Theorem and theTwo-Series Theorem
	3.2 Proofs
		3.2.1 The Gradient Lemma
		3.2.2 The Estimate of Var(SN)
		3.2.3 McLeish\'s Martingale Central Limit Theorem
		3.2.4 Proof of the Central Limit Theorem
		3.2.5 Convergence of Moments
		3.2.6 Characterization of Center-Tight Additive Functionals
		3.2.7 Proof of the Two-Series Theorem
	*3.3 The Almost Sure Invariance Principle
	3.4 Notes and References
4 The Essential Range and Irreducibility
	4.1 Definitions and Motivation
	4.2 Main Results
		4.2.1 Markov Chains
		4.2.2 Markov Arrays
		4.2.3 Hereditary Arrays
	4.3 Proofs
		4.3.1 Reduction Lemma
		4.3.2 Joint Reduction
		4.3.3 The Possible Values of the Co-Range
		4.3.4 Calculation of the Essential Range
		4.3.5 Existence of Irreducible Reductions
		4.3.6 Characterization of Hereditary Additive Functionals
	4.4 Notes and References
5 The Local Limit Theorem in the Irreducible Case
	5.1 Main Results
		5.1.1 Local Limit Theorems for Markov Chains
		5.1.2 Local Limit Theorems for Markov Arrays
		5.1.3 Mixing Local Limit Theorems
	5.2 Proofs
		5.2.1 Strategy of Proof
		5.2.2 Characteristic Function Estimates
		5.2.3 The LLT via Weak Convergence of Measures
		5.2.4 The LLT in the Irreducible Non-Lattice Case
		5.2.5 The LLT in the Irreducible Lattice Case
		5.2.6 Mixing LLT
	5.3 Notes and References
6 The Local Limit Theorem in the Reducible Case
	6.1 Main Results
		6.1.1 Heuristics and Warm Up Examples
		6.1.2 The LLT in the Reducible Case
		6.1.3 Irreducibility as a Necessary Condition for the Mixing LLT
		6.1.4 Universal Bounds for Prob[SN-zN(a,b)]
	6.2 Proofs
		6.2.1 Characteristic Functions in the Reducible Case
		6.2.2 Proof of the LLT in the Reducible Case
		6.2.3 Necessity of the Irreducibility Assumption
		6.2.4 Universal Bounds for Markov Chains
		6.2.5 Universal Bounds for Markov Arrays
	6.3 Notes and References
7 Local Limit Theorems for Moderate Deviationsand Large Deviations
	7.1 Moderate Deviations and Large Deviations
	7.2 Local Limit Theorems for Large Deviations
		7.2.1 The Log Moment Generating Functions
		7.2.2 The Rate Functions
		7.2.3 The LLT for Moderate Deviations
		7.2.4 The LLT for Large Deviations
	7.3 Proofs
		7.3.1 Strategy of Proof
		7.3.2 A Parameterized Family of Changes of Measure
		7.3.3 Choosing the Parameters
		7.3.4 The Asymptotic Behavior of V\"0365VξN(SN)
		7.3.5 Asymptotics of the Log Moment Generating Functions
		7.3.6 Asymptotics of the Rate Functions
		7.3.7 Proof of the Local Limit Theorem for Large Deviations
		7.3.8 Rough Bounds in the Reducible Case
	7.4 Large Deviations Thresholds
		7.4.1 The Large Deviations Threshold Theorem
		7.4.2 Admissible Sequences
		7.4.3 Proof of the Large Deviations Threshold Theorem
		7.4.4 Examples
	7.5 Notes and References
8 Important Examples and Special Cases
	8.1 Introduction
	8.2 Sums of Independent Random Variables
	8.3 Homogenous Markov Chains
	*8.4 One-Step Homogeneous Additive Functionals in L2
	8.5 Asymptotically Homogeneous Markov Chains
	8.6 Equicontinuous Additive Functionals
	8.7 Notes and References
9 Local Limit Theorems for Markov Chains in RandomEnvironments
	9.1 Markov Chains in Random Environments
		9.1.1 Formal Definitions
		9.1.2 Examples
		9.1.3 Conditions and Assumptions
	9.2 Main Results
	9.3 Proofs
		9.3.1 Existence of Stationary Measures
		9.3.2 The Essential Range is Almost Surely Constant
		9.3.3 Variance Growth
		9.3.4 Irreducibility and the LLT
		9.3.5 LLT for Large Deviations
	9.4 Notes and References
A The Gärtner-Ellis Theorem in One Dimension
	A.1 The Statement
	A.2 Background from Convex Analysis
	A.3 Proof of the Gärtner-Ellis Theorem
	A.4 Notes and References
B Hilbert\'s Projective Metric and Birkhoff\'s Theorem
	B.1 Hilbert\'s Projective Metric
	B.2 Contraction Properties
	B.3 Notes and References
C Perturbations of Operators with Spectral Gap
	C.1 The Perturbation Theorem
	C.2 Some Facts from Analysis
	C.3 Proof of the Perturbation Theorem
	C.4 Notes and References
References
Index