Geometric sums: bounds for rare events with applications: risk analysis, reliability, queueing
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1997
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| Series: | Mathematics and its applications
413 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 265 S. graph. Darst. |
| ISBN: | 0792346165 |
Staff View
MARC
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| 245 | 1 | 0 | |a Geometric sums: bounds for rare events with applications |b risk analysis, reliability, queueing |c by Vladimir Kalashnikov |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1997 | |
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Record in the Search Index
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| adam_text | Table of Contents
Preface xi
Glossary of Notation xvi
Chapter 1. Introduction 1
1 General Remarks 1
2 Geometric and Exponential Distributions 4
3 Origins of Geometric Sums 9
3.1 Miscellaneous 9
3.2 Random Walk 12
3.3 Queueing Models 15
3.4 Thinning of Random Flows 18
3.5 Reliability Models 19
3.6 Risk Theory 20
4 Background Material 23
4.1 General Setup 23
4.2 Discussion 26
Commentaries 28
Chapter 2. Miscellaneous Probability Topics 30
1 Probability Metrics 30
1.1 Definition 30
1.2 Examples 31
1.3 Properties 34
1.4 Relationships between metrics 37
2 Classes of Probability Distributions 40
2.1 Uniformly Integrable Family 40
vii
viii Table of Contents
2.2 Uniformly Weighted Family 44
2.3 Heavy Tailed Distributions 47
3 Moments of Sums of I.I.D.R.V. s 51
4 Renewal Processes 54
5 Markov Chains 61
6 Ladder Heights 63
Commentaries 71
Chapter 3. Generalized Renyi Theorem 74
1 Geometric Sums and Renewal Theory 74
2 The Renyi Limit Theorem 76
2.1 Motivation 76
2.2 Summands with a Finite Exponential Moment 78
2.3 Uniformly Weighted Summands 82
2.4 Delayed Geometric Sums 86
3 Heavy Traffic in Queueing 88
4 General Markov Chains 90
4.1 Introductory Remarks and Notation 90
4.2 Splitting 92
4.3 Rare Excursions 94
4.4 Limiting Results 97
Commentaries 99 ;
Chapter 4. Two Sided Bounds 101
1 The Cramer Case 101
2 Exponential Bounds in Queueing 105
3 Heavy Tailed Summands 107
3.1 Upper Bounds 107
3.2 Lower Bounds 113
3.3 Lower Bounds by Test Functions 123
3.4 Asymptotic Behaviour 127
3.5 Discussion 128
4 Examples 129
Commentaries 143
Chapter 5. Metric Bounds 146
1 Motivation 146
2 Preliminary Results 147
3 Bounds in Terms of the Mean Metric 150
Table of Contents ix
4 Bounds in Terms of the Uniform Metric 152
5 Continuity of Geometric Sums 158
6 Geometric Sums with Alternating Signs Summands 160
7 Multivariate Summands 164
Commentaries 169
Chapter 6. Ruin Probability 171
1 Basic Notation and Problems 171
2 The Initial Capital Securing a Prescribed Risk 175
3 Two Sided Bounds for Ruin Probabilities 179
3.1 General Remarks 179
3.2 The Cramer Case 183
3.3 The Heavy Tailed Case 190
4 Continuity of Ruin Probability 195
4.1 Classical Risk Model 195
3.2 The S. Andersen Risk Model 198
Commentaries 199
Chapter 7. Reliability Regenerative Models 201
1 Preliminaries 201
2 Basic definitions 202
3 The First Occurrence Time and Geometric Sums 205
4 Semi regenerative Models 219
4.1 Basic Constructions 219
4.2 The First Occurrence Time 223
4.3 Successive Approximations 227
4.4 Integral Equations 229
4.5 Monte Carlo Algorithms 241
Commentaries 247
References 250
Index 259
|
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| id | DE-604.BV011677684 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:13:52Z |
| institution | BVB |
| isbn | 0792346165 |
| language | English |
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| physical | XVII, 265 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Kalašnikov, Vladimir Vjačeslavovič 1942-2001 Verfasser (DE-588)1013229029 aut Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing by Vladimir Kalashnikov Dordrecht [u.a.] Kluwer 1997 XVII, 265 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 413 Stochastischer Prozess - Zuverlässigkeitstheorie Geometrische Verteilung (DE-588)4507247-4 gnd rswk-swf Unabhängige Zufallsvariable (DE-588)4205983-5 gnd rswk-swf Geometrische Verteilung (DE-588)4507247-4 s Unabhängige Zufallsvariable (DE-588)4205983-5 s DE-604 Mathematics and its applications 413 (DE-604)BV008163334 413 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007872446&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kalašnikov, Vladimir Vjačeslavovič 1942-2001 Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing Mathematics and its applications Stochastischer Prozess - Zuverlässigkeitstheorie Geometrische Verteilung (DE-588)4507247-4 gnd Unabhängige Zufallsvariable (DE-588)4205983-5 gnd |
| subject_GND | (DE-588)4507247-4 (DE-588)4205983-5 |
| title | Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing |
| title_auth | Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing |
| title_exact_search | Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing |
| title_full | Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing by Vladimir Kalashnikov |
| title_fullStr | Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing by Vladimir Kalashnikov |
| title_full_unstemmed | Geometric sums: bounds for rare events with applications risk analysis, reliability, queueing by Vladimir Kalashnikov |
| title_short | Geometric sums: bounds for rare events with applications |
| title_sort | geometric sums bounds for rare events with applications risk analysis reliability queueing |
| title_sub | risk analysis, reliability, queueing |
| topic | Stochastischer Prozess - Zuverlässigkeitstheorie Geometrische Verteilung (DE-588)4507247-4 gnd Unabhängige Zufallsvariable (DE-588)4205983-5 gnd |
| topic_facet | Stochastischer Prozess - Zuverlässigkeitstheorie Geometrische Verteilung Unabhängige Zufallsvariable |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007872446&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT kalasnikovvladimirvjaceslavovic geometricsumsboundsforrareeventswithapplicationsriskanalysisreliabilityqueueing |