Cycle Representations of Markov Processes (eBook)

Sophia L. Kalpazidou (Autor)

eBook Download: PDF

2007 | 2nd ed. 2006
XX, 304 Seiten
Springer New York (Verlag)
978-0-387-36081-2 (ISBN)

Infos

"This book is a prototype providing new insight into Markovian dependence via the cycle decompositions. It presents a systematic account of a class of stochastic processes known as cycle (or circuit) processes - so-called because they may be defined by directed cycles. These processes have special and important properties through the interaction between the geometric properties of the trajectories and the algebraic characterization of the Markov process. An important application of this approach is the insight it provides to electrical networks and the duality principle of networks. In particular, it provides an entirely new approach to infinite electrical networks and their applications in topics as diverse as random walks, the classification of Riemann surfaces, and to operator theory. The second edition of this book adds new advances to many directions, which reveal wide-ranging interpretations of the cycle representations like homologic decompositions, orthogonality equations, Fourier series, semigroup equations, and disintegration of measures. The versatility of these interpretations is consequently motivated by the existence of algebraic-topological principles in the fundamentals of the cycle representations. This book contains chapter summaries as well as a number of detailed illustrations. Review of the earlier edition: ""This is a very useful monograph which avoids ready ways and opens new research perspectives. It will certainly stimulate further work, especially on the interplay of algebraic and geometrical aspects of Markovian dependence and its generalizations."" Math Reviews."

Inhalt

Preface to the Second Edition 7
Preface 9
Acknowledgments 15
Contents 16
Fundamentals of the Cycle Representations of Markov Processes 20
Directed Circuits 21
1.1 Definition of Directed Circuits 22
1.2 The Passage Functions 26
1.3 Cycle Generating Equations 28
Genesis of Markov Chains by Circuits: The Circuit Chains 35
2.1 Finite Markov Chains Defined by Weighted Circuits 35
2.2 Denumerable Markov Chains Generated by Circuits 41
Cycle Representations of Recurrent Denumerable Markov Chains 47
3.1 The Derived Chain of Qians 47
3.2 The Circulation Distribution of a Markov Chain 53
3.3 A Probabilistic Cycle Decomposition for Recurrent Markov Chains 55
3.4 Weak Convergence of Sequences of Circuit Chains: A Deterministic Approach 57
3.5 Weak Convergence of Sequences of Circuit Chains: A Probabilistic Approach 63
3.6 The Induced Circuit Chain 65
Circuit Representations of Finite Recurrent Markov Chains 72
4.1 Circuit Representations by Probabilistic Algorithms 73
4.2 Circuit Representations by Nonrandomized Algorithms 74
4.3 The Caratheodory-Type Circuit Representations 77
4.4 The Betti Number of a Markov Chain 78
4.5 A Refined Cycle Decomposition of Finite Stochastic Matrices: A Homologic Approach 83
4.6 The Dimensions of Caratheodory and Betti 89
Continuous Parameter Circuit Processes with Finite State Space 90
5.1 Genesis of Markov Processes by Weighted Circuits 90
5.2 The Weight Functions 93
5.3 Continuity Properties of the Weight Functions 96
5.4 Differentiability Properties of the Weight Functions 100
5.5 Cycle Representation Theorem for Transition Matrix Functions 102
5.6 Cycle Representation Theorem for Q-Matrices 105
Spectral Theory of Circuit Processes 109
6.1 Unitary Dilations in Terms of Circuits 109
6.2 Integral Representations of the Circuit-Weights Decomposing Stochastic Matrices 112
6.3 Spectral Representation of Continuous Parameter Circuit Processes 114
Higher-Order Circuit Processes 117
7.1 Higher-Order Markov Chains 117
7.2 Higher-Order Finite Markov Chains Defined by Weighted Circuits 122
7.3 The Rolling-Circuits 133
7.4 The Passage-Function Associated with a Rolling- Circuit 136
7.5 Representation of Finite Multiple Markov Chains by Weighted Circuits 139
Cycloid Markov Processes 147
8.1 The Passages Through a Cycloid 147
8.2 The Cycloid Decomposition of Balanced Functions 151
8.3 The Cycloid Transition Equations 153
8.4 Definition of Markov Chains by Cycloids 157
Markov Processes on Banach Spaces on Cycles 160
9.1 Banach Spaces on Cycles 160
9.2 Fourier Series on Directed Cycles 167
9.3 Orthogonal Cycle Transforms for Finite Stochastic Matrices 172
9.4 Denumerable Markov Chains on Banach Spaces on Cycles 176
The Cycle Measures 178
10.1 The Passage-Functions as Characteristic Functions 178
10.2 The Passage-Functions as Balanced Functions 182
10.3 The Vector Space Generated by the Passage- Functions 186
10.4 The Cycle Measures 189
10.5 Measures on the Product of Two Measurable Spaces by Cycle Representations of Balanced Functions: A Fubini- Type Theorem 196
Wide-Ranging Interpretations of the Cycle Representations of Markov Processes 201
11.1 The Homologic Interpretation of the Cycle Processes 201
11.2 An Algebraic Interpretation 206
11.3 The Banach Space Approach 208
11.4 The Measure Theoretic Interpretation 209
11.5 The Cycle Representation Formula as a Disintegration of Measures 211
Applications of the Cycle Representations 218
Stochastic Properties in Terms of Circuits 219
1.1 Recurrence Criterion in Terms of the Circuits 219
1.2 The Entropy Production of Markov Chains 222
1.3 Reversibility Criteria in Terms of the Circuits 224
1.4 Derriennic Recurrence Criterions in Terms of the Weighted Circuits 227
Levy’s Theorem Concerning Positiveness of Transition Probabilities 236
2.1 Levy’s Theorem in Terms of Circuits 237
2.2 Physical Interpretation of the Weighted Circuits Representing a Markov Process 239
The Rotational Theory of Markov Processes 242
3.1 Preliminaries 242
3.2 Joel E. Cohen’s Conjecture on Rotational Representations of Stochastic Matrices 245
3.3 Alpern’s Solution to the Rotational Problem 246
3.4 Transforming Circuits into Circle Arcs 251
3.5 Mapping Stochastic Matrices into Partitions and a Probabilistic Solution to the Rotational Problem 258
3.6 The Rotational Dimension of Stochastic Matrices and a Homologic Solution to the Rotational Problem 261
3.7 The Complexity of the Rotational Representations 266
3.8 A Reversibility Criterion in Terms of Rotational Representations 270
3.9 Rotational Representations of Transition Matrix Functions 273
List of Notations 276
Bibliography 278
Index 308

Produkt-Details

• Erscheint lt. Verlag: 6.3.2007
• Reihe/Serie: Stochastic Modelling and Applied Probability
• Reihe/Serie: Stochastic Modelling and Applied Probability
• Zusatzinfo: XX, 304 p. 17 illus.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Markov Chain • Markov process • Markov Processes • Random Walk • Stochastic process • Stochastic Processes
• ISBN-10: 0-387-36081-6 / 0387360816
• ISBN-13: 978-0-387-36081-2 / 9780387360812