Conditional measures and applications

M.M. Rao1574445936, 9781574445930, 9781420027433

In response to unanswered difficulties in the generalized case of conditional expectation and to treat the topic in a well-deservedly thorough manner, M.M. Rao gave us the highly successful first edition of Conditional Measures and Applications. Until this groundbreaking work, conditional probability was relegated to scattered journal articles and mere chapters in larger works on probability. This second edition continues to offer a thorough treatment of conditioning while adding substantial new information on developments and applications that have emerged over the past decade. Conditional Measures and Applications, Second Edition clearly elucidates the subject, from fundamental principles to abstract analysis. The author illustrates the computational difficulties in evaluating conditional probabilities in nondiscrete cases with numerous examples, demonstrates applications to Markov processes, martingales, potential theory, and Reynolds operators as well as sufficiency in statistics, and clarifies ideas in modern noncommutative probability structures through conditioning in general structures, including parts of operator algebras and free random variables. He also discusses existence and construction problems from the Bishop-Brouwer constructive analysis point of view. With open problems in every chapter and links to other areas of mathematics, this invaluable second edition offers complete coverage of conditional probability and expectation and their structural analysis, from simple to advanced abstract levels, for both novices and seasoned mathematicians.

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Table of contents :
dk3825fm.pdf……Page 1
Conditional Measures and Applications, Second Edition……Page 4
Preface to the Second Edition……Page 7
Preface to the First Edition……Page 12
Contents……Page 17
References……Page 0
1.1 Introduction……Page 21
1.2. Conditional probability given a partition……Page 23
1.3 Conditional expectation: elementary case……Page 30
1.4 Conditioning with densities……Page 33
1.5 Conditional probability spaces: first steps……Page 38
1.6 Bibliographical notes……Page 43
2.1 Introduction of the general concept……Page 45
2.2 Basic properties of conditional expectations……Page 52
2.3 Conditional probabilities in the general case……Page 63
2.4 Remarks on the inclusion of previous concepts……Page 71
2.5 Conditional independence and related concepts……Page 74
2.6 Bibliographical notes……Page 81
3.1 Introduction……Page 84
3.2 Some examples with multiple solutions: paradoxes……Page 85
3.3 Dissection of paradoxes……Page 96
3.4 Some methods of computation……Page 99
3.5 Remarks on traditional calculations of conditional measures……Page 107
3.6 Bibliographical notes……Page 111
4.1 Introduction……Page 113
4.2 Axiomatization of conditioning based on partitions……Page 115
4.3 Structure of the new conditional probability functions……Page 119
4.4 Some applications……Page 127
4.5 Difficulties with earlier examples persist……Page 136
4.6 Bibliographical notes……Page 142
5.1 Introduction……Page 145
5.2 Existence of regular conditional probabilities: generalities……Page 146
5.3 Special spaces admitting regular conditional probabilities……Page 151
5.4 Disintegration of probability measures and regular conditioning……Page 164
5.5 Further results on disintegration……Page 179
5.6 Evaluation of conditional expectations by Fourier analysis……Page 184
5.7 Further evaluations of conditional expectations……Page 193
5.8 Bibliographical notes……Page 200
6.1. Introduction……Page 203
6.2 Conditioning relative to families of measures……Page 207
6.3 Sufficiency: the dominated case……Page 210
6.4 Sufficiency: the undominated case……Page 219
6.5 Sufficiency: another approach to the undominated case……Page 231
6.6 Bibliographical notes……Page 237
7.1 Introduction……Page 239
7.2 Integration relative to conditional measures and function spaces……Page 241
7.3 Functional characterizations of conditioning……Page 255
7.4 Integral representations of conditional expectations……Page 276
7.5 Renyi’s formulation as a specialization of the abstract version……Page 295
7.6 Conditional measures and differentiation……Page 299
7.7 Bibliographical notes……Page 304
8.1 Introduction……Page 306
8.2 A general formulation of products……Page 312
8.3 General projective limit theorems……Page 318
8.4 Some consequences……Page 330
8.5 Remarks on conditioning, disintegration, and lifting……Page 337
8.6 Bibliographical notes……Page 342
9.1 Introduction……Page 344
9.2 Set martingales……Page 350
9.3 Martingale convergence……Page 359
9.4 Markov processes: some basic results……Page 370
9.5 Further properties of Markov processes……Page 381
9.6 Bibliographical notes……Page 386
10.1 Introduction and motivation……Page 388
10.2 Conditional measures and potential kernels……Page 392
10.3 Reynolds operators and conditional expectations……Page 402
10.4 Bistochastic operators and conditioning……Page 404
10.5 Contractive projections and conditional expectations……Page 412
10.6 Bibliographical notes……Page 419
11.1 Introduction……Page 422
11.2 Averagings in cones of positive functions……Page 424
11.3 Averaging operators on function algebras……Page 427
11.4 Conditioning in operator algebras……Page 430
11.5 Free independence and a bijection in operator algebras……Page 444
11.6 Some applications of noncommutative conditioning……Page 461
11.7 Bibliographical notes……Page 472
References……Page 474
Notations……Page 490