# Martingale Limit Theory and Its Application

## 1st Edition

Authors:
Editors:
eBook ISBN: 9781483263229
Published Date: 28th December 1980
Page Count: 320
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## Description

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.

﻿Preface

Notation

1 Introduction

1.1. General Definition

1.2. Historical Interlude

1.3. The Martingale Convergence Theorem

1.4. Comments on Classical Limit Theory and Its Analogs

1.5. On the Repertoire of Available Limit Theory

1.6. Martingale Limit Theorems Generalizing Those for Sums of Independent Random Variables

1.7. Martingale Limit Theorems Viewed as Rate of Convergence Results in the Martingale Convergence Theorem

2 Inequalities and Laws of Large Numbers

2.1. Introduction

2.2. Basic Inequalities

2.3. The Martingale Convergence Theorem

2.4. Square Function Inequalities

2.5. Weak Law of Large Numbers

2.6. Strong Law of Large Numbers

2.7. Convergence in LP

3 The Central Limit Theorem

3.1. Introduction

3.2. The Central Limit Theorem

3.3. Toward a General Central Limit Theorem

3.4. Raikov-Type Results in the Martingale CLT

3.5. Reverse Martingales and Martingale Tail Sums

3.6. Rates of Convergence in the CLT

4 Invariance Principles in the Central Limit Theorem and Law of the Iterated Logarithm

4.1. Introduction

4.2. Invariance Principles in CLT

4.3. Rates of Convergence for the Invariance Principle in the CLT

4.4. The Law of the Iterated Logarithm and Its Invariance Principle

5 Limit Theory for Stationary Processes via Corresponding Results for Approximating Martingales

5.1. Introduction

5.2. The Probabilistic Framework

5.3. The Central Limit Theorem

5.4. Functional Forms of the Central Limit Theorem and Law of the Iterated Logarithm

5.5. The Central Limit Theorem via Approximation to the Distribution of the Stationary Process

6 Estimation of Parameters from Stochastic Processes

6.1. Introduction

6.2. Asymptotic Behaviour of the Maximum Likelihood Estimator

6.3. Conditional Least Squares

6.4. Quadratic Functions of Discrete Time Series

6.5. The Method of Moments

7 Miscellaneous Applications

7.1. Exchangeable Sequences

7.2. Limit Laws for Subsequences of Sequences of Random Variables

7.3. Limit Laws for Subadditive Processes

7.4. The Hawkins Random Sieve

7.5. Genetic Balance When the Population Size Is Varying

7.6. Stochastic Approximation

7.7. On Adaptive Control of Linear Systems

Appendix

I. The Skorokhod Representation

II. Weak Convergence on Function Spaces

III. Mixing Inequalities

IV. Stationarity and Ergodicity

V. Miscellanea

References

Index to Theorems, Corollaries, and Examples

Index

## Details

No. of pages:
320
Language:
English
Published:
28th December 1980
Imprint:
eBook ISBN:
9781483263229