Cycle representations of Markov processes / Sophia L. Kalpazidou.

Format:

Book

Author/Creator:

Kalpazidou, Sophia L.

Contributor:

Hazel M. Hussong Fund.

Series:

Stochastic modelling and applied probability 0172-4568 ; 28.

Stochastic modelling and applied probability, 0172-4568 ; 28

Language:

English

Subjects (All):

Markov processes.

Algebraic cycles.

Physical Description:

xx, 301 pages : illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

New York : Springer, [2006]

Summary:

This book presents an original and 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 finite-dimensional distributions. An important application of this approach is the new insight it provides into Markovian dependence and electrical networks. In particular, it provides an entirely new approach to Markov processes and infinite electrical networks, and their applications in topics as diverse as random walks, ergodic theory, dynamical systems, potential theory, theory of matrices, algebraic topology, complexity theory, the classification of Riemann surfaces, and operator theory.

The author surveys the three principal developments in cycle theory: the cycle-decomposition formula and its relation to the Markov process; entropy production and how it may be used to measure how far a process is from being reversible; and how a finite recurrent stochastic matrix may be defined by a rotation of the circle and a partition whose elements consist of finite unions of circle-arcs. The cycle representations have been advanced after the publication of the first edition to many directions, which reveal wide-ranging interpretations like homologic decompositions, orthogonality equations, Fourier series, semigroup equations, disintegration of measures, and so on. The versatility of these interpretations is consequently motivated by the existence of algebraic-topological principles in the fundamentals of the cycle representations, which elaborates the standard view on the Markovian modelling to new intuitive and constructive approaches.

Contents:

I Fundamentals of the Cycle Representations of Markov Processes

1 Directed Circuits 3

1.1 Definition of Directed Circuits 4

1.2 The Passage Functions 8

1.3 Cycle Generating Equations 10

2 Genesis of Markov Chains by Circuits: The Circuit Chains 17

2.1 Finite Markov Chains Defined by Weighted Circuits 17

2.2 Denumerable Markov Chains Generated by Circuits 23

3 Cycle Representations of Recurrent Denumerable Markov Chains 29

3.1 The Derived Chain of Qians 29

3.2 The Circulation Distribution of a Markov Chain 35

3.3 A Probabilistic Cycle Decomposition for Recurrent Markov Chains 37

3.4 Weak Convergence of Sequences of Circuit Chains: A Deterministic Approach 39

3.5 Weak Convergence of Sequences of Circuit Chains: A Probabilistic Approach 45

3.6 The Induced Circuit Chain 47

4 Circuit Representations of Finite Recurrent Markov Chains 55

4.1 Circuit Representations by Probabilistic Algorithms 56

4.2 Circuit Representations by Nonrandomized Algorithms 57

4.3 The Caratheodory-Type Circuit Representations 60

4.4 The Betti Number of a Markov Chain 61

4.5 A Refined Cycle Decomposition of Finite Stochastic Matrices: A Homologic Approach 66

4.6 The Dimensions of Caratheodory and Betti 72

5 Continuous Parameter Circuit Processes with Finite State Space 73

5.1 Genesis of Markov Processes by Weighted Circuits 73

5.2 The Weight Functions 76

5.3 Continuity Properties of the Weight Functions 79

5.4 Differentiability Properties of the Weight Functions 83

5.5 Cycle Representation Theorem for Transition Matrix Functions 85

5.6 Cycle Representation Theorem for Q-Matrices 88

6 Spectral Theory of Circuit Processes 93

6.1 Unitary Dilations in Terms of Circuits 93

6.2 Integral Representations of the Circuit-Weights Decomposing Stochastic Matrices 96

6.3 Spectral Representation of Continuous Parameter Circuit Processes 98

7 Higher-Order Circuit Processes 101

7.1 Higher-Order Markov Chains 101

7.2 Higher-Order Finite Markov Chains Defined by Weighted Circuits 106

7.3 The Rolling-Circuits 117

7.4 The Passage-Function Associated with a Rolling-Circuit 120

7.5 Representation of Finite Multiple Markov Chains by Weighted Circuits 122

8 Cycloid Markov Processes 131

8.1 The Passages Through a Cycloid 131

8.2 The Cycloid Decomposition of Balanced Functions 135

8.3 The Cycloid Transition Equations 138

8.4 Definition of Markov Chains by Cycloids 141

9 Markov Processes on Banach Spaces on Cycles 145

9.1 Banach Spaces on Cycles 145

9.2 Fourier Series on Directed Cycles 152

9.3 Orthogonal Cycle Transforms for Finite Stochastic Matrices 157

9.4 Denumerable Markov Chains on Banach Spaces on Cycles 161

10 The Cycle Measures 163

10.1 The Passage-Functions as Characteristic Functions 163

10.2 The Passage-Functions as Balanced Functions 167

10.3 The Vector Space Generated by the Passage-Functions 171

10.4 The Cycle Measures 175

10.5 Measures on the Product of Two Measurable Spaces by Cycle Representations of Balanced Functions: A Fubini-Type Theorem 182

11 Wide-Ranging Interpretations of the Cycle Representations of Markov Processes 187

11.1 The Homologic Interpretation of the Cycle Processes 187

11.2 An Algebraic Interpretation 192

11.3 The Banach Space Approach 194

11.4 The Measure Theoretic Interpretation 195

11.5 The Cycle Representation Formula as a Disintegration of Measures 197

II Applications of the Cycle Representations

1 Stochastic Properties in Terms of Circuits 207

1.1 Recurrence Criterion in Terms of the Circuits 207

1.2 The Entropy Production of Markov Chains 210

1.3 Reversibility Criteria in Terms of the Circuits 212

1.4 Derriennic Recurrence Criterions in Terms of the Weighted Circuits 215

2 Levy's Theorem Concerning Positiveness of Transition Probabilities 225

2.1 Levy's Theorem in Terms of Circuits 226

2.2 Physical Interpretation of the Weighted Circuits Representing a Markov Process 228

3 The Rotational Theory of Markov Processes 231

3.2 Joel E. Cohen's Conjecture on Rotational Representations of Stochastic Matrices 234

3.3 Alpern's Solution to the Rotational Problem 235

3.4 Transforming Circuits into Circle Arcs 240

3.5 Mapping Stochastic Matrices into Partitions and a Probabilistic Solution to the Rotational Problem 247

3.6 The Rotational Dimension of Stochastic Matrices and a Homologic Solution to the Rotational Problem 250

3.7 The Complexity of the Rotational Representations 255

3.8 A Reversibility Criterion in Terms of Rotational Representations 259

3.9 Rotational Representations of Transition Matrix Functions 262.

Notes:

Includes bibliographical references (pages [267]-296) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Hazel M. Hussong Fund.

ISBN:

0387291660

OCLC:

63514198