Stochastic limit theory : an introduction for econometricians / James Davidson.

Format:

Book

Author/Creator:

Davidson, James, 1944- author.

Language:

English

Subjects (All):

Econometrics.

Limit theorems (Probability theory).

Stochastic processes.

Physical Description:

1 online resource (808 pages)

Edition:

Second edition.

Place of Publication:

Oxford, United Kingdom : Oxford University Press, 2021.

Summary:

Stochastic Limit Theory has become a standard reference in its field. This new edition offers updated and improved results and an extended range of topics. It works both as a textbook and as an account of recent work in a field of particular interest to econometricians.

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.

Notes:

Description based on print version record.

ISBN:

0-19-192720-1

0-19-265880-8

OCLC:

1283845969