Stopping Times and Directed Processes (Encyclopedia of Mathematics and its Applications, Series Number 47)

The notion of 'stopping times' is a useful one in probability theory; it can be applied to both classical problems and fresh ones. This book presents this technique in the context of the directed set, stochastic processes indexed by directed sets, and many applications in probability, analysis and ergodic theory. Martingales and related processes are considered from several points of view. The book opens with a discussion of pointwise and stochastic convergence of processes, with concise proofs arising from the method of stochastic convergence. Later, the rewording of Vitali covering conditions in terms of stopping times clarifies connections with the theory of stochastic processes. Solutions are presented here for nearly all the open problems in the Krickeberg convergence theory for martingales and submartingales indexed by directed set. Another theme of the book is the unification of martingale and ergodic theorems. In This Book The Technique Of Stopping Times Is Applied To Prove Convergence Theorems For Stochastic Processes - In Particular Processes Indexed By Direct Sets - And In Sequential Analysis. Applications Of Convergence Theorems Are Seen In Probability, Analysis, And Ergodic Theory. Almost Everywhere, Convergence And Stochastic Convergence Of Processes Indexed By A Directed Set Are Studied, And Solutions Are Given For Problems Left Open In Krickeberg's Theory For Martingales And Submartingales. The Rewording Of Vitali Covering Conditions In Terms Of Stopping Times Establishes Connections With The Theory Of Stochastic Processes And Derivation. A Study Of Martingales Yields Laws Of Large Numbers For Martingale Differences, With Application To Star-mixing Processes. Convergence Of Processes Taking Values In Banach Spaces Is Related To Geometric Properties Of These Spaces. There Is A Self-contained Section On Operator Ergodic Theorems: The Superadditive, Chacon-ornstein, And Chacon Theorems. A Recurrent Theme Of The Book Is The Unification Of Martingale And Ergodic Theorems. One Example Is The Use Of A Three-function Inequality, Which Is Basic In All The One And Many Parameter Results. A General Principle Is Proved Showing That In Both Theories All The Multiparameter Convergence Theorems Follow From One-parameter Maximal And Convergence Theorems. Requiring Only A Knowledge Of Basic Measure Theory, This Book Will Be A Valuable Reference For Students And Researchers In Probability Theory, Analysis, And Statistics.--book Jacket. 1. Stopping Times. 1.1. Definitions. 1.2. The Amart Convergence Theorem. 1.3. Directed Processes And The Radon-nikodym Theorem. 1.4. Conditional Expectations -- 2. Infinite Measure And Orlicz Spaces. 2.1. Orlicz Spaces. 2.2. More On Orlicz Spaces. 2.3. Uniform Integrability And Conditional Expectation -- 3. Inequalities. 3.1. The Three-function Inequality. 3.2. Sharp Maximal Inequality For Martingale Transforms. 3.3. Prophet Compared To Gambler -- 4. Directed Index Set. 4.1. Essential And Stochastic Convergence. 4.2. The Covering Condition (v). 4.3. L[suscript Psi]-bounded Martingales. 4.4. L[subscript 1]-bounded Martingales -- 5. Banach-valued Random Variables. 5.1. Vector Measures And Integrals. 5.2. Martingales And Amarts. 5.3. The Radon-nikodym Property. 5.4. Geometric Properties. 5.5. Operator Ideals -- 6. Martingales. 6.1. Maximal Inequalities For Supermartingales. 6.2. Decompositions Of Submartingales. 6.3. The Norm Of The Square Function Of A Martingale. 6.4. Lifting -- 7. Derivation. 7.1. Derivation In [actual Symbols Not Reproducible]. 7.2. Derivation In [actual Symbols Not Reproducible]. 7.3. Abstract Derivation. 7.4. D-bases -- 8. Pointwise Ergodic Theorems. 8.1. Preliminaries. 8.2. Weak Maximal Inequalities. 8.3. Hopf's Decomposition. 8.4. The [sigma]-algebra Of Absorbing Sets. 8.5. The Chacon-ornstein Theorem (conservative Case). 8.6. Superadditive Processes -- 9. Multiparameter Processes. 9.1. A Multiparameter Convergence Principle. 9.2. Multiparameter Cesaro Averages Of Operators. 9.3. Multiparameter Ratio Ergodic Theorems. 9.4. Multiparameter Martingales. G.a. Edgar And Louis Sucheston. Includes Bibliographical References And Index. Title ......Page 5 Copyright ......Page 6 Contents ......Page 7 Preface ......Page 11 1. Stopping times ......Page 15 1.1. Definitions ......Page 16 1.2. The amatt convergence theorem ......Page 23 1.3. Directed processes and the Radon-Nikodym theorem ......Page 27 1.4. Conditional expectations ......Page 33 2. Infinite measure and Orlicz spaces ......Page 47 2.1. Orlicz spaces ......Page 48 2.2. More on Orlicz spaces ......Page 65 2.3. Uniform integrability and conditional expectation ......Page 82 3.1. The three-function inequality ......Page 96 3.2. Sharp maximal inequality for martingale transforms ......Page 113 3.3. Prophet compared to gambler ......Page 117 4. Directed index set ......Page 127 4.1. Essential and stochastic convergence ......Page 128 4.2. The covering condition (V) ......Page 141 4.3. Lip-bounded martingales ......Page 158 4.4. Ll-bounded martingales ......Page 173 5. Banach-valued random variables ......Page 185 5.1. Vector measures and integrals ......Page 186 5.2. Martingales and amarts ......Page 197 5.3. The Radon-Nikodym property ......Page 212 5.4. Geometric properties ......Page 232 5.5. Operator ideals ......Page 246 6.1. Maximal inequalities for supermartingales ......Page 268 6.2. Decompositions of submartingales ......Page 281 6.3. The norm of the square function of a martingale ......Page 287 6.4. Lifting ......Page 294 7.1. Derivation in R ......Page 305 7.2. Derivation in Rd ......Page 314 7.3. Abstract derivation ......Page 322 7.4. D-bases ......Page 341 8. Pointwise ergodic theorems ......Page 358 8.1. Preliminaries ......Page 359 8.2. Weak maximal inequalities ......Page 364 8.3. Hopf's decomposition ......Page 369 8.4. The a-algebra of absorbing sets ......Page 370 8.5. The Chacon-Ornstein theorem (conservative case) ......Page 375 8.6. Superadditive processes ......Page 379 9. Multiparameter processes ......Page 396 9.1. A multiparameter convergence principle ......Page 397 9.2. Multiparameter Cesaro averages of operators ......Page 404 9.3. Multiparameter ratio ergodic theorems ......Page 408 9.4. Multiparameter martingales ......Page 410 References ......Page 421 Index of names ......Page 432 Index of terms ......Page 435 Cover ......Page 443

The notion of "stopping times" is a useful one in probability theory; it can be applied to both classical problems and new ones. This book presents this technique in the context of the directed set, stochastic processes indexed by directed sets, and provides many applications in probability, analysis, and ergodic theory. The book opens with a discussion of pointwise and stochastic convergence of processes with concise proofs arising from the method of stochastic convergence. Later, the rewording of Vitali covering conditions in terms of stopping times, clarifies connections with the theory of stochastic processes. Solutions are presented here for nearly all the open problems in the Krickeberg convergence theory for martingales and submartingales indexed by directed set. Another theme is the unification of martingale and ergodic theorems. Among the topics treated are: the three-function maximal inequality, Burkholder's martingale transform inequality and prophet inequalities, convergence in Banach spaces, and a general superadditive ration ergodic theorem. From this, the general Chacon-Ornstein theorem and the Chacon theorem can be derived. A second instance of the unity of ergodic and martingale theory is a general principle showing that in both theories, all the multiparameter convergence theorems follow from one-parameter maximal and convergence theorems.