Hall / Heyde / Birnbaum Martingale Limit Theory and Its Application

1. Auflage 2014
ISBN: 978-1-4832-6322-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 320 Seiten, Web PDF

Martingale Limit Theory and Its Application discusses the asymptotic properties of martingales, particularly as regards key prototype of probabilistic behavior that has wide applications. The book explains the thesis that martingale theory is central to probability theory, and also examines the relationships between martingales and processes embeddable in or approximated by Brownian motion. The text reviews the martingale convergence theorem, the classical limit theory and analogs, and the martingale limit theorems viewed as the rate of convergence results in the martingale convergence theorem. The book explains the square function inequalities, weak law of large numbers, as well as the strong law of large numbers. The text discusses the reverse martingales, martingale tail sums, the invariance principles in the central limit theorem, and also the law of the iterated logarithm. The book investigates the limit theory for stationary processes via corresponding results for approximating martingales and the estimation of parameters from stochastic processes. The text can be profitably used as a reference for mathematicians, advanced students, and professors of higher mathematics or statistics.

Autoren/Hrsg.

Hall, P.

Heyde, C. C.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Martingale Limit Theory and its Application;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Notation;13
7;Chapter 1. Introduction;14
7.1;1.1. General Definition;14
7.2;1.2. Historical Interlude;14
7.3;1.3. The Martingale Convergence Theorem;15
7.4;1.4. Comments on Classical Limit Theory and Its Analogs;16
7.5;1.5. On the Repertoire of Avaiiabie Limit Theory;19
7.6;1.6. Martingale Limit Theorems Generalizing Those for Sums of Independent Random Variables;21
7.7;1.7. Martingale Limit Theorems Viewed as Rate of Convergence Results in the Martingale Convergence Theorem;23
8;Chapter 2. Inequalities and Laws ofLarge Numbers;26
8.1;2.1. Introduction;26
8.2;2.2. Basic Inequaiities;27
8.3;2.3 The Martingale Convergence Theorem;30
8.4;2.4. Square Function Inequalities;36
8.5;2.5. Weak Law of Large Numbers;42
8.6;2.6. Strong Law of Large Numbers;44
8.7;2.7 Convergence in LP;54
9;Chapter 3. The Central Limit Theorem;64
9.1;3.1. Introduction;64
9.2;3.2. The Central Limit Theorem;65
9.3;3.3. Toward a General Central Limit Theorem;78
9.4;3.4. Raikov-Type Results in the Martingale CLT;82
9.5;3.5. Reverse Martingales and Martingale Tail Sums;89
9.6;3.6. Rates off Convergence in the CLT;94
10;Chapter 4. Invariance Principles in the Central Limit Theorem and Law of the Iterated Logarithm;110
10.1;4.1. Introduction;110
10.2;4.2. Invariance Principles in the CLT;110
10.3;4.3. Rates of Convergence for the Invariance Principle in the CLT;122
10.4;4.4. The Law of the Iterated Logarithm and its Invarlance Principle;128
11;Chapter 5. Limit Theory for Stationary Processes via Corresponding Results for Approximating Martingales;140
11.1;5.1. Introduction;140
11.2;5.2. The Probabilistic Framework;141
11.3;5.3. The Central Limit Theorem;141
11.4;5.4. Functional Forms of the Central Limit Theorem and Law of the Iterated Logarithm;153
11.5;5.5. The Central Limit Theorem via Approximation to the Distribution of the Stationary Process;160
12;Chapter 6. Estimation of Parameters from Stochastic Processes;168
12.1;6.1. Introduction;168
12.2;6.2. Asymptotic Behaviour of the Maximum Likelihood Estimator;169
12.3;6.3. Conditional Least Squares;185
12.4;6.4. Quadratic Functions of Discrete Time Series;195
12.5;6.5. The Method of Moments;208
13;Chapter 7. Miscellaneous Applications;214
13.1;7.1. Exchangeable Sequences;214
13.2;7.2. Limit Laws for Subsequences of Sequences of Random Variables;218
13.3;7.3. Limit Laws for Subadditive Processes;226
13.4;7.4. The Hawkins Random Sieve;239
13.5;7.5. Genetic Balance When the Population Size is Varying;248
13.6;7.6. Stochastic Approximation;251
13.7;7.7. On Adaptive Control of Linear Systems;265
14;Appendix;282
14.1;I. The Skorokhod Representation;282
14.2;II. Weak Convergence on Function Spaces;286
14.3;III. Mixing Inequalities;289
14.4;IV. Stationarity and Ergodicity;293
14.5;V. Miscellanea;294
15;References;298
16;Index to T'heorem, and Examples;312
17;Index;314