Bellow / Jones Almost Everywhere Convergence II

Proceedings of the International Conference on Almost Everywhere Convergence in Probability and Ergodic Theory, Evanston, Illinois, October 16-20, 1989
1. Auflage 2014
ISBN: 978-1-4832-6592-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 288 Seiten, Web PDF

Almost Everywhere Convergence II presents the proceedings of the Second International Conference on Almost Everywhere Convergence in Probability and Ergodotic Theory, held in Evanston, Illinois on October 16-20, 1989. This book discusses the many remarkable developments in almost everywhere convergence. Organized into 19 chapters, this compilation of papers begins with an overview of a generalization of the almost sure central limit theorem as it relates to logarithmic density. This text then discusses Hopf's ergodic theorem for particles with different velocities. Other chapters consider the notion of a log-convex set of random variables, and proved a general almost sure convergence theorem for sequences of log-convex sets. This book discusses as well the maximal inequalities and rearrangements, showing the connections between harmonic analysis and ergodic theory. The final chapter deals with the similarities of the proofs of ergodic and martingale theorems. This book is a valuable resource for mathematicians.

Autoren/Hrsg.

Bellow, Alexandra

Jones, Roger L.

Inhaltsverzeichnis

1;Front Cover;1
2;Almost Everywhere Convergence II;4
3;Copyright Page;5
4;Table of Contents;6
5;CONTRIBUTORS;8
6;CONFERENCE PARTICIPANTS;10
7;Dedication;12
8;Preface;14
9;Chapter 1. A Solution to a Problem of A. Bellow;16
9.1;References;22
10;Chapter 2. Universal Weights from Dynamical Systems To Mean–Bounded Positive Operators on Lp;24
10.1;Introduction;25
10.2;Definition;25
10.3;References;31
11;Chapter 3. SOME CONNECTIONS BETWEEN ERGODIC THEORY AND HARMONIC ANALYSIS;32
11.1;1. Introduction;32
11.2;2. Equivalences and implications among several maximal estimates;34
11.3;3. More maximal operators, weak (1, 1), Wiener-Wintner, and spectral continuity;38
11.4;REFERENCES;53
12;Chapter 4. On Hopfs Ergodic Theorem for Particles with Different Velocities;56
12.1;Summary;56
12.2;1. Hopfs Billiard;56
12.3;2. A Counterexample;57
12.4;3. The Local Ergodic Theorem;60
12.5;References;62
13;Chapter 5. A Note on the Strong Law of Large Numbers for Partial Sums of Independent Random Vectors;64
13.1;1. Introduction and Preparatory Results;64
13.2;2. Stability Results for Vector Valued Random Variables;69
13.3;3. Possible Extensions;77
13.4;4. Proof of Theorem 2.1;79
13.5;References;82
14;Chapter 6. SUMMABILITY METHODS AND ALMOST-SURE CONVERGENCE;84
14.1;0. Introduction;84
14.2;1. Limits of occupation times;85
14.3;2. Cesaro and Riesz means;91
14.4;3. Euler, Borel and related methods;92
14.5;4. Complements;94
14.6;REFERENCES;95
15;Chapter 7. Concerning Induced Operators and Alternating Sequences;100
15.1;1 Introduction;100
15.2;2 Lp Convergence of Alternating Sequences;102
15.3;3 An L1 counterexample;106
15.4;References;106
16;Chapter 8. Maximal inequalities and ergodic theorems for Cesàro-a or weighted averages;108
16.1;1) The strong law of large numbers and the ergodic theorem for Cesàro-a averages : a résumé;108
16.2;2) Maximal inequality and ergodic theorem for weighted averages : a general formulation;111
16.3;3) The "precise" form of the maximal inequality;116
16.4;REFERENCES;121
17;Chapter 9. THE HILBERT TRANSFORM OF THE GAUSSIAN;124
17.1;BIBLIOGRAPHY;127
18;Chapter 10. Mean Ergodicity of L1 Contractions and Pointwise Ergodic Theorems;128
18.1;0 Introduction;128
18.2;1 On Ergodic Convergence for a General L1 Contraction;130
18.3;2 Ergodic Convergence of Commuting L1 Contractions;134
18.4;References;140
19;Chapter 11. Multi–Parameter Moving Averages;142
19.1;Introduction;142
19.2;Section I: Averages over Cubes;144
19.3;Proof of Theorem 1.1. part a);148
19.4;Proof of Theorem 1.1. part b);150
19.5;Section 2: Averages over Rectangles;155
19.6;Proof of Theorem 2.1 part b;157
19.7;Section 3: Strong Sweeping Out;159
19.8;References;163
20;Chapter 12. An Almost Sure Convergence Theorem For Sequences of Random Variables Selected From Log-Convex Sets;166
20.1;1. Log-Convex Sets of Random Variables;167
20.2;2. Derivation of Main Result;172
20.3;3. Examples;179
20.4;References;181
21;Chapter 13. DIVERGENCE OF ERGODIC AVERAGES AND ORBITAL CLASSIFICATION OF NON-SINGULAR TRANSFORMATIONS;182
21.1;Definitions and conventions;183
21.2;REFERENCES;193
22;Chapter 14. SOME ALMOST SURE CONVERGENCE PROPERTIES OF WEIGHTEDSUMS OF MARTINGALE DIFFERENCE SEQUENCES;194
22.1;1. Introduction;194
22.2;2. Matrix-normed strong laws;195
22.3;3. Triangular arrays of nonrandom weights;198
22.4;REFERENCES;205
23;Chapter 15. Pointwise ergodic theorems for certain order preserving mappings in L1†;206
23.1;1. INTRODUCTION;206
23.2;2. ORDER PRESERVING MAPPINGS IN L1;207
23.3;3. THE Sn ITERATION;210
23.4;5. POINTWISE ERGODIC THEOREMS;213
23.5;6. GENERALIZED MEASURE PRESERVING TRANSFORMATIONS;217
23.6;REFERENCES;222
24;Chapter 16. On the almost sure central limit theorem;224
24.1;1. Introduction and results;225
24.2;2. Proofs;230
24.3;References;239
25;Chapter 17. UNIVERSALLY BAD SEQUENCES IN ERGODIC THEORY;242
25.1;1. Definition;243
25.2;2. Definition;244
25.3;3. Definition;245
25.4;4. Definition;246
25.5;5. Proposition;246
25.6;6. Corollary;247
25.7;7. Question;247
25.8;8. Theorem;247
25.9;9. Remark;247
25.10;10. Proposition;248
25.11;11. Corollary;248
25.12;12. Remark.;248
25.13;13. Proposition;248
25.14;14. Remark;249
25.15;15. Proposition;249
25.16;16. Theorem;250
25.17;17. Remark;251
25.18;18. Proposition;251
25.19;19. Remark;251
25.20;20. Theorem;251
25.21;21. Lemma;254
25.22;22. Remark;254
25.23;References;259
26;Chapter 18. On an Inequality of Kahane;262
26.1;References;266
27;Chapter 19. A PRINCIPLE FOR ALMOST EVERYWHERE CONVERGENCE OF MULTIPARAMETER PROCESSES;268
27.1;Introduction;268
27.2;1. A Multiparameter Convergence Principle;269
27.3;2. Convergence of multiparameter martingales;276
27.4;3. A multiparameter ratio ergodic theorem;282
27.5;References;286