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El. knyga: Ergodicity of Markov Processes via Nonstandard Analysis

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Ergodicity of Markov Processes via Nonstandard AnalysisDidesnis paveikslėlis
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"The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logicand nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes"--

The Markov chain ergodic theorem is well understood if either the time-line or the state space is discrete, say Duanmu, Rosenthal, and Weiss, but there is not a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, they introduce a class of hyperfinite general Markov processes that behave like finite state space discrete-time Markov processes. They show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)
Chapter 1 Introduction
1(6)
1.1
Chapter Outline
2(5)
Chapter 2 Markov Processes and the Main Result
7(8)
Chapter 3 Preliminaries: Nonstandard Analysis
15(8)
3.1 The Hyperreals
18(2)
3.2 Nonstandard Extensions of General Metric Spaces
20(3)
Chapter 4 Internal Probability Theory
23(6)
4.1 Product Measures
24(2)
4.2 Nonstandard Integration Theory
26(3)
Chapter 5 Measurability of Standard Part Map
29(4)
Chapter 6 Hyperfinite Representation of a Probability Space
33(6)
Chapter 7 General Hyperfinite Markov Processes
39(12)
Chapter 8 Hyperfinite Representation for Discrete-time Markov Processes
51(10)
8.1 General properties of the transition probability
51(2)
8.2 Hyperfinite Representation for Discrete-time Markov Processes
53(8)
Chapter 9 Hyperfinite Representation for Continuous-time Markov Processes
61(16)
9.1 Construction of Hyperfinite State Space
62(4)
9.2 Construction of Hyperfinite Markov Processess
66(11)
Chapter 10 Markov Chain Ergodic Theorem
77(8)
Chapter 11 The Feller Condition
85(10)
11.1 Hyperfinite Representation under the Feller Condition
86(5)
11.2 A Weaker Markov Chain Ergodic Theorem
91(4)
Chapter 12 Push-down Results
95(8)
12.1 Construction of Standard Markov Processes
96(3)
12.2 Push down of Weakly Stationary Distributions
99(2)
12.3 Existence of Stationary Distributions
101(2)
Chapter 13 Merging of Markov Processes
103(4)
Chapter 14 Miscellaneous Remarks
107(3)
Acknowledgement 110(3)
Bibliography 113
Haosui Duanmu, University of Toronto, ON, Canada.

Jeffrey S. Rosenthal, University of Toronto, ON, Canada.

William Weiss, University of Toronto, ON, Canada.
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