Stochastic limit theory : an introduction for econometricians

Cover
Stochastic Limit Theory: An Introduction for Econometricians
Copyright
Dedication
Contents
From Preface to the First Edition
Preface to the Second Edition
Mathematical Symbols and Abbreviations
	Common Usages
Part I: Mathematics
	1: Sets and Numbers
		1.1 Basic Set Theory
		1.2 Mappings
		1.3 Countable Sets
		1.4 The Real Continuum
		1.5 Sequences of Sets
		1.6 Classes of Subsets
		1.7 Sigma Fields
		1.8 The Topology of the Real Line
	2: Limits, Sequences, and Sums
		2.1 Sequences and Limits
		2.2 Functions and Continuity
		2.3 Vector Sequences and Functions
		2.4 Sequences of Functions
		2.5 Summability and Order Relations
		2.6 Inequalities
		2.7 Regular Variation
		2.8 Arrays
	3: Measure
		3.1 Measure Spaces
		3.2 The Extension Theorem
		3.3 Non-measurability
		3.4 Product Spaces
		3.5 Measurable Transformations
		3.6 Borel Functions
	4: Integration
		4.1 Construction of the Integral
		4.2 Properties of the Integral
		4.3 Product Measure and Multiple Integrals
		4.4 The Radon–Nikodym Theorem
	5: Metric Spaces
		5.1 Spaces
		5.2 Distances and Metrics
		5.3 Separability and Completeness
		5.4 Examples
		5.5 Mappings on Metric Spaces
		5.6 Function Spaces
	6: Topology
		6.1 Topological Spaces
		6.2 Countability and Compactness
		6.3 Separation Properties
		6.4 Weak Topologies
		6.5 The Topology of Product Spaces
		6.6 Embedding and Metrization
Part II: Probability
	7: Probability Spaces
		7.1 Probability Measures
		7.2 Conditional Probability
		7.3 Independence
		7.4 Product Spaces
	8: Random Variables
		8.1 Measures on the Line
		8.2 Distribution Functions
		8.3 Examples
		8.4 Multivariate Distributions
		8.5 Independent Random Variables
	9: Expectations
		9.1 Averages and Integrals
		9.2 Applications
		9.3 Expectations of Functions of X
		9.4 Moments
		9.5 Theorems for the Probabilist’s Toolbox
		9.6 Multivariate Distributions
		9.7 More Theorems for the Toolbox
		9.8 Random Variables Depending on a Parameter
	10: Conditioning
		10.1 Conditioning in Product Measures
		10.2 Conditioning on a Sigma Field
		10.3 Conditional Expectations
		10.4 Some Theorems on Conditional Expectations
		10.5 Relationships between Sub-?-fields
		10.6 Conditional Distributions
	11: Characteristic Functions
		11.1 The Distribution of Sums of Random Variables
		11.2 Complex Numbers
		11.3 The Theory of Characteristic Functions
		11.4 Examples
		11.5 Infinite Divisibility
		11.6 The Inversion Theorem
		11.7 The Conditional Characteristic Function
Part III: Theory of Stochastic Processes
	12: Stochastic Processes
		12.1 Basic Ideas and Terminology
		12.2 Convergence of Stochastic Sequences
		12.3 The Probability Model
		12.4 The Consistency Theorem
		12.5 Uniform and Limiting Properties
		12.6 Uniform Integrability
	13: Time Series Models
		13.1 Independence and Stationarity
		13.2 The Poisson Process
		13.3 Linear Processes
		13.4 Random Walks
	14: Dependence
		14.1 Shift Transformations
		14.2 Invariant Events
		14.3 Ergodicity and Mixing
		14.4 Sub-?-fields and Regularity
		14.5 Strong and Uniform Mixing
	15: Mixing
		15.1 Mixing Sequences of Random Variables
		15.2 Mixing Inequalities
		15.3 Mixing in Linear Processes
		15.4 Sufficient Conditions for Strong and Uniform Mixing
	16: Martingales
		16.1 Sequential Conditioning
		16.2 Extensions of the Martingale Concept
		16.3 Martingale Convergence
		16.4 Convergence and the Conditional Variances
		16.5 Martingale Inequalities
	17: Mixingales
		17.1 Definition and Examples
		17.2 Telescoping Sum Representations
		17.3 Maximal Inequalities
		17.4 Uniform Square-Integrability
		17.5 Autocovariances
	18: Near-Epoch Dependence
		18.1 Definitions and Examples
		18.2 Near-Epoch Dependence and Mixingales
		18.3 Transformations
		18.4 Adaptation
		18.5 Approximability
		18.6 NED in Volatility
Part IV: The Law of Large Numbers
	19: Stochastic Convergence
		19.1 Almost Sure Convergence
		19.2 Convergence in Probability
		19.3 Transformations and Convergence
		19.4 Convergence in Lp Norm
		19.5 Examples
		19.6 Laws of Large Numbers
	20: Convergence in Lp Norm
		20.1 Weak Laws by Mean Square Convergence
		20.2 Almost Sure Convergence by the Method of Subsequences
		20.3 Truncation Arguments
		20.4 A Martingale Weak Law
		20.5 Mixingale Weak Laws
		20.6 Approximable Processes
	21: The Strong Law of Large Numbers
		21.1 Technical Tricks for Proving LLNs
		21.2 The Case of Independence
		21.3 Martingale Strong Laws
		21.4 Conditional Variances and Random Weighting
		21.5 Strong Laws for Mixingales
		21.6 NED and Mixing Processes
	22: Uniform Stochastic Convergence
		22.1 Stochastic Functions on a Parameter Space
		22.2 Pointwise and Uniform Convergence
		22.3 Stochastic Equicontinuity
		22.4 Generic Uniform Convergence
		22.5 Uniform Laws of Large Numbers
Part V: The Central Limit Theorem
	23: Weak Convergence of Distributions
		23.1 Basic Concepts
		23.2 The Skorokhod Representation Theorem
		23.3 Weak Convergence and Transformations
		23.4 Convergence of Moments and Characteristic Functions
		23.5 Criteria for Weak Convergence
		23.6 Convergence of Random Sums
		23.7 Stable Distributions
	24: The Classical Central Limit Theorem
		24.1 The I.I.D. Case
		24.2 Independent Heterogeneous Sequences
		24.3 Feller’s Theorem and Asymptotic Negligibility
		24.4 The Case of Trending Variances
		24.5 Gaussianity by Other Means
		24.6 ?-Stable Convergence
	25: CLTs for Dependent Processes
		25.1 A General Convergence Theorem
		25.2 The Martingale Case
		25.3 Stationary Ergodic Sequences
		25.4 The CLT for Mixingales
		25.5 NED Functions of Mixing Processes
	26: Extensions and Complements
		26.1 The CLT with Estimated Normalization
		26.2 The CLT for Linear Processes
		26.3 The CLT with Random Norming
		26.4 The Multivariate CLT
		26.5 The Delta Method
		26.6 Law of the Iterated Logarithm
		26.7 Berry–Esséen Bounds
Part VI: The Functional Central Limit Theorem
	27: Measures on Metric Spaces
		27.1 Separability and Measurability
		27.2 Measures and Expectations
		27.3 Function Spaces
		27.4 The Space C
		27.5 Measures on C
		27.6 Wiener Measure
	28: Stochastic Processes in Continuous Time
		28.1 Adapted Processes
		28.2 Diffusions and Martingales
		28.3 Brownian Motion
		28.4 Properties of Brownian Motion
		28.5 Skorokhod Embedding
		28.6 Processes Derived from Brownian Motion
		28.7 Independent Increments and Continuity
	29: Weak Convergence
		29.1 Weak Convergence in Metric Spaces
		29.2 Skorokhod’s Representation
		29.3 Metrizing the Space of Measures
		29.4 Tightness and Convergence
		29.5 Weak Convergence in C
		29.6 An FCLT for Martingale Differences
		29.7 The Multivariate Case
	30: Càdlàg Functions
		30.1 The Space D
		30.2 Metrizing D
		30.3 Billingsley’s Metric
		30.4 Measures on D
		30.5 Prokhorov’s Metric
		30.6 Compactness and Tightness in D
		30.7 Weak Convergence in D
	31: FCLTs for Dependent Variables
		31.1 Asymptotic Independence
		31.2 NED Functions of Mixing Processes 1
		31.3 NED Functions of Mixing Processes 2
		31.4 Nonstationary Increments
		31.5 Generalized Partial Sums
		31.6 The Multivariate Case
	32: Weak Convergence to Stochastic Integrals
		32.1 Weak Limit Results for Random Functionals
		32.2 Stochastic Integrals
		32.3 Convergence to Stochastic Integrals
		32.4 Convergence in Probability to ???
Bibliography
Index