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Viser: An Introduction to Markov Chains 2017
An Introduction to Markov Chains 2017
Anders Tolver
(2017)
Sprog: Engelsk
Department of Mathematical Sciences
160,00 kr.
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- 191 sider
- Udgiver: Department of Mathematical Sciences (November 2017)
- ISBN: 9788770788991
These lecture notes have been developed for the course Stochastic Processes at Department of Mathematical Sciences, University of Copenhagen during the teaching years 2010-2016. The material covers aspects of the theory for time-homogeneous Markov chains in discrete and continuous time on finite or countable state spaces.
Introduction 7
Motivation and some examples of Markov chains 7
About these lecture notes 11
Transition diagrams 14
Overview of exercises 16
2 Markov chains in discrete time 17
Definition of a Markov chain 17
Classification of states 22
Limit results and invariant probabilities 35
Absorption probabilities 44
Exercises 57
3 Markov chains in continuous time 79
Definition and the minimal construction of a Markov chain 79
Properties of the transition probabilities 85
Invariant probabilities and absorption 92
Birth-and-death processes 107
Exercises 117
A Random variables and stochastic processes 149
Probability measures 149
Random variables 150
Stochastic processes 152
B Mathematical tools 157
B.1 Elementary conditional probabilities 157
B.2 Some formulaes for sums and series 160
B.3 Some results for matrices 161
B.4 First order differential equations 164
B.5 Second order linear recurrence equations 164
B.6 The ratio test 165
B.7 Integral test for convergence 166
B.8 How to do certain computations in R 167
C Proofs of selected results 77
C.1 Recurrence criterion 1 177
C.2 Number of visits to state j 178
C.3 Invariant distribution 180
C.4 Uniqueness of invariant distribution 183
C.5 On the ergodic theorem for discrete-time Markov chains 184
D Bibliography 189
E Index 191
Motivation and some examples of Markov chains 7
About these lecture notes 11
Transition diagrams 14
Overview of exercises 16
2 Markov chains in discrete time 17
Definition of a Markov chain 17
Classification of states 22
Limit results and invariant probabilities 35
Absorption probabilities 44
Exercises 57
3 Markov chains in continuous time 79
Definition and the minimal construction of a Markov chain 79
Properties of the transition probabilities 85
Invariant probabilities and absorption 92
Birth-and-death processes 107
Exercises 117
A Random variables and stochastic processes 149
Probability measures 149
Random variables 150
Stochastic processes 152
B Mathematical tools 157
B.1 Elementary conditional probabilities 157
B.2 Some formulaes for sums and series 160
B.3 Some results for matrices 161
B.4 First order differential equations 164
B.5 Second order linear recurrence equations 164
B.6 The ratio test 165
B.7 Integral test for convergence 166
B.8 How to do certain computations in R 167
C Proofs of selected results 77
C.1 Recurrence criterion 1 177
C.2 Number of visits to state j 178
C.3 Invariant distribution 180
C.4 Uniqueness of invariant distribution 183
C.5 On the ergodic theorem for discrete-time Markov chains 184
D Bibliography 189
E Index 191
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