http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Cycle representations of Markov processes
1 online resource (xv, 194 pages) : illustrations.
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.
Directed Circuits. Definition of Directed Circuits. The Passage Functions. Cycle Generating Equations -- Genesis of Markov Chains by Circuits: The Circuit Chains. Finite Markov Chains Defined by Weighted Circuits. Denumerable Markov Chains Generated by Circuits -- Cycle Representations of Recurrent Denumerable Markov Chains. The Derived Chain of Qians. The Circulation Distribution of a Markov Chain. A Probabilistic Cycle Decomposition for Recurrent Markov Chains. Weak Convergence of Sequences of Circuit Chains: A Deterministic Approach. Weak Convergence of Sequences of Circuit Chains: A Probabilistic Approach -- Circuit Representations of Finite Recurrent Markov Chains. Circuit Representations by Probabilistic Algorithms. Circuit Representations by Nonrandomized Algorithms. The Caratheodory-Type Circuit Representations. The Betti Number of a Markov Chain. A Refined Cycle Decomposition of Finite Stochastic Matrices: A Homologic Approach. The Dimensions of Caratheodory and Betti.
Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002. http://purl.oclc.org/DLF/benchrepro0212 MiAaHDL
digitized 2010 HathiTrust Digital Library committed to preserve pda MiAaHDL
Includes bibliographical references (pages 169-190) and index.
9780387943633 (New York : acid-free paper)
9783540943631 (Berlin : acid-free paper)