Markov chain models: rarity and exponentiality
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, [u.a.]
Springer
[1979]
|
| Series: | Applied mathematical sciences
Volume 28 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiii, 184 Seiten |
| ISBN: | 0387904050 3540904050 |
Staff View
MARC
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| 245 | 1 | 0 | |a Markov chain models |b rarity and exponentiality |c Julian Keilson |
| 264 | 1 | |a New York, [u.a.] |b Springer |c [1979] | |
| 264 | 4 | |c © 1979 | |
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Record in the Search Index
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Contents
Page
FOREWORD AND ACKNOWLEDGMENT v
NOTATION vii
CHAPTER 0. INTRODUCTION AND SUMMARY 1
CHAPTER 1. DISCRETE TIME MARKOV CHAINS; REVERSIBILITY
IN TIME 15
§1.00. Introduction 15
§1.0. Notation, Transition Laws 15
§1.1. Irreducibility, Aperiodicity,
Ergodicity; Stationary Chains 16
§1.2. Approach to Ergodicity; Spectral
Structure, Perron Romanovsky
Frobenius Theorem 17
§1.3. Time Reversible Chains 18
CHAPTER 2. MARKOV CHAINS IN CONTINUOUS TIME; UNIFORMIZA
TION; REVERSIBILITY 20
§2.00. Introduction 20
§2.0. Notation, Transition Laws; A Review. 20
§2.1. Uniformizable Chains A Bridge
Between Discrete and Continuous Time
Chains 2 2
§2.2. Advantages and Prevalence of Uni¬
formizable Chains 24
§2.3. Ergodicity for Continuous Time
Chains 2 5
§2.4. Reversibility for Ergodic Markov
Chains in Continuous Time 26
§2.5. Prevalence of Time Reversibility. 27
CHAPTER 3. MORE ON TIME REVERSIBILITY; POTENTIAL COEFF¬
ICIENTS; PROCESS MODIFICATION 31
§3.00. Introduction 31
§3.1. The Advantages of Time Reversibility 32
§3.2. The Spectral Representation 32
ix
X
Page
5 3.3. Potentials; Spectral Representa¬
tion 35
§3.4. More General Time Reversible Chains. 38
§3.5. Process Modifications Preserving
Reversibility 38
§3.6. Replacement Processes 41
CHAPTER 4. POTENTIAL THEORY, REPLACEMENT, AND COMPENSA¬
TION 43
§4.00. Introduction 43
§4.1. The Green Potential 44
§4.2. The Ergodic Distribution for a Re¬
placement Process 45
§4.3. The Compensation Method 47
§4.4. Notation for the Homogeneous Random
Walk 47
§4.5. The Compensation Method Applied to
the Homogeneous Random Walk Modified
by Boundaries 49
§4.6. Advantages of the Compensation
Method. An Illustrative Example 51
§4.7. Exploitation of the Structure of the
Green Potential for the Homogeneous
Random Walk 53
§4.8. Similar Situations 56
CHAPTER 5. PASSAGE TIME DENSITIES IN BIRTH DEATH
PROCESSES ; DISTRIBUTION STRUCTURE 57
§5.00. Introduction 57
§5.1. Passage Time Densities for Birth
Death Processes 57
§5.2. Passage Time Moments for a Birth
Death Process 61
§5.3. PFoo, Complete Monotonicity, Log
Concavity and Log Convexity 63
§5.4. Complete Monotonicity and Log
Convexity 66
§5.5. Complete Monotonicity in Time
Reversible Processes 67
xi
Page
55.6. Some Useful Inequalities for the
Families CM and PF^ 68
§5.7. Log Concavity and Strong Unimodality
for Lattice Distributions 70
§5.8. Preservation of Log Concavity and
Log Convexity under Tail Summation
and Integration 73
§5.9. Relation of CM and PF«, to IFR and
DFR Classes in Reliability 74
CHAPTER 6. PASSAGE TIMES AND EXIT TIMES FOR MORE
GENERAL CHAINS 76
§6.00. Introduction 76
§6.1. Passage Time Densities to a Set of
States 77
§6.2. Mean Passage Times to a Set via the
Green Potential 81
§6.3. Ruin Probabilities via the Green
Potential 84
§6.4. Ergodic Flow Rates in a Chain. 86
§6.5. Ergodic Exit Times, Ergodic Sojourn
Times, and Quasi Stationary Exit
Times 88
§6.6. The Quasi Stationary Exit Time.
A Limit Theorem 90
§6.7. The Connection Between Exit Times
and Sojourn Times. A Renewal Theorem 92
§6.8. A Comparison of the Mean Ergodic
Exit Time and Mean Ergodic Sojourn
Time for Arbitrary Chains 97
§6.9. Stochastic Ordering of Exit Times of
Interest for Time Reversible Chains. 99
§6.10. Superiority of the Exit Time as
System Failure Time; Jitter 102
xii
Page
CHAPTER 7. THE FUNDAMENTAL MATRIX, AND ALLIED TOPICS. 105
§7.00. Introduction 105
§7.1. The Fundamental Matrix for Ergodic
Chains 106
§7.2. The Structure of the Fundamental
Matrix for Time Reversible Chains. 109
§7.3. Mean Failure Times and Ruin Proba¬
bilities for Systems with Indepen¬
dent Markov Components and More
General Chains 112
§7.4. Covariance and Spectral Density
Structure for Time Reversible
Processes 118
§7.5. A Central Limit Theorem 121
§7.6. Regeneration Times and Passage Times
Their Relation For Arbitrary Chains. 122
§7.7. Passage to a Set with Two States. 125
CHAPTER 8. RARITY AND EXPONENTIALITY 130
§8.0. Introduction 130
§8.1. Passage Time Density Structure for
Finite Ergodic Chains; the Exponen¬
tial Approximation 131
§8.2. A Limit Theorem for Ergodic Regener¬
ative Processes 133
§8.3. Prototype Behavior: Birth Death
Processes; Strongly Stable Systems. 137
§8.4. Limiting Behavior of the Ergodic and
Quasi stationary Exit Time Densities
and Sojourn Time Densities for Birth
Death Processes 143
§8.5. Limit Behavior of Other Exit Times
for More General Chains 145
§8.6. Strongly Stable Chains, Jitter;
Estimation of the Failure Time
Needed for the Exponential Approxi¬
mation 150
§8.7. A Measure of Exponentiality in the
Completely Monotone Class of
Densities 152
xiii
Page
§8.8. An Error Bound for Departure from
Exponentiality in the Completely
Monotone Class 155
§8.9. The Exponential Approximation for
Time Reversible Systems 156
§8.10. A Relaxation Time of Interest 161
CHAPTER 9. STOCHASTIC MONOTONICITY 164
§9.00. Introduction 164
§9.1. Monotone Markov Matrices and Mono¬
tone Chains 164
§9.2. Some Monotone Chains in Discrete
Time 168
§9.3. Monotone Chains in Continuous Time. 171
§9.4. Other Monotone Processes in Continu¬
ous Time 174
REFERENCES 176
INDEX 181 |
| any_adam_object | 1 |
| author | Keilson, Julian 1924-1999 |
| author_GND | (DE-588)1073848361 |
| author_facet | Keilson, Julian 1924-1999 |
| author_role | aut |
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| author_variant | j k jk |
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| classification_rvk | QH 237 SK 820 SK 800 |
| ctrlnum | (OCoLC)4774537 (DE-599)BVBBV002258671 |
| dewey-full | 519.2/33 510/.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics 510 - Mathematics |
| dewey-raw | 519.2/33 510/.8 |
| dewey-search | 519.2/33 510/.8 |
| dewey-sort | 3519.2 233 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| isbn | 0387904050 3540904050 |
| language | English |
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| physical | xiii, 184 Seiten |
| publishDate | 1979 |
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| publisher | Springer |
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| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Keilson, Julian 1924-1999 (DE-588)1073848361 aut Markov chain models rarity and exponentiality Julian Keilson New York, [u.a.] Springer [1979] © 1979 xiii, 184 Seiten txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences Volume 28 Markov, Processus de Markov, processus de ram Markov processes Markov-Modell (DE-588)4168923-9 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Markov-Kette (DE-588)4037612-6 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 s DE-604 Markov-Kette (DE-588)4037612-6 s Markov-Modell (DE-588)4168923-9 s Erscheint auch als Online-Ausgabe 978-1-4612-6200-8 Applied mathematical sciences Volume 28 (DE-604)BV000005274 28 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001484304&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Keilson, Julian 1924-1999 Markov chain models rarity and exponentiality Applied mathematical sciences Markov, Processus de Markov, processus de ram Markov processes Markov-Modell (DE-588)4168923-9 gnd Markov-Prozess (DE-588)4134948-9 gnd Markov-Kette (DE-588)4037612-6 gnd |
| subject_GND | (DE-588)4168923-9 (DE-588)4134948-9 (DE-588)4037612-6 |
| title | Markov chain models rarity and exponentiality |
| title_auth | Markov chain models rarity and exponentiality |
| title_exact_search | Markov chain models rarity and exponentiality |
| title_full | Markov chain models rarity and exponentiality Julian Keilson |
| title_fullStr | Markov chain models rarity and exponentiality Julian Keilson |
| title_full_unstemmed | Markov chain models rarity and exponentiality Julian Keilson |
| title_short | Markov chain models |
| title_sort | markov chain models rarity and exponentiality |
| title_sub | rarity and exponentiality |
| topic | Markov, Processus de Markov, processus de ram Markov processes Markov-Modell (DE-588)4168923-9 gnd Markov-Prozess (DE-588)4134948-9 gnd Markov-Kette (DE-588)4037612-6 gnd |
| topic_facet | Markov, Processus de Markov, processus de Markov processes Markov-Modell Markov-Prozess Markov-Kette |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001484304&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT keilsonjulian markovchainmodelsrarityandexponentiality |