Stochastic limit theory : an introduction for econometricians /

'Stochastic Limit Theory', published in 1994, has become a standard reference in its field. Now reissued in a new edition, offering updated and improved results and an extended range of topics, Davidson surveys asymptotic (large-sample) distribution theory with applications to econometrics...

Full description

Bibliographic Details
Main Author:	Davidson, James, 1944- (Author)
Format:	eBook
Language:	English
Published:	Oxford : Oxford University Press, 2021.
Edition:	Second edition.
Subjects:	Econometrics. Limit theorems (Probability theory) Stochastic processes. Stochastic Processes Économétrie. Théorèmes limites (Théorie des probabilités) Processus stochastiques. Econometrics Stochastic processes Electronic books.
Online Access:	Connect to the full text of this electronic book

Table of Contents:

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.