Lectures on the Poisson process:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2018
|
| Schriftenreihe: | Institute of Mathematical Statistics textbooks
7 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xx, 293 Seiten Illustrationen |
| ISBN: | 9781107088016 9781107458437 |
Internformat
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Datensatz im Suchindex
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| adam_text | Contents page xv xix Preface List of Symbols 1 1.1 1.2 1.3 1.4 1.5 Poisson and Other Discrete Distributions The Poisson Distribution Relationships Between Poisson and Binomial Distributions The Poisson Limit Theorem The Negative Binomial Distribution Exercises 2 2.1 2.2 2.3 2.4 2.5 Point Processes Fundamentals Campbell’s Formula Distribution of a Point Process Point Processes on Metric Spaces Exercises 3 3.1 3.2 3.3 3.4 Poisson Processes Definition of the Poisson Process Existence of Poisson Processes Laplace Functional of the Poisson Process Exercises 4 4.1 4.2 4.3 4.4 4.5 The Mecke Equation and FactorialMeasures The Mecke Equation Factorial Measures and the Multivariate Mecke Equation Janossy Measures Factorial Moment Measures Exercises ix 1 1 3 4 5 7 9 9 12 14 16 18 19 19 20 23 24 26 26 28 32 34 36
Contents x 5 5.1 5.2 5.3 5.4 Mappings, Markings and Thinnings 6 Characterisations of the Poisson Process 6.1 6.2 6.3 6.4 6.5 6.6 Borel Spaces Simple Point Processes Rényi’s Theorem Completely Orthogonal Point Processes Turning Distributional into Almost Sure Identities Exercises 7 Poisson Processes on the Real Line 7.1 7.2 7.3 7.4 7.5 The Interval Theorem Marked Poisson Processes Record Processes Polar Representation of Homogeneous Poisson Processes Exercises 8 Stationary Point Processes 8.1 8.2 8.3 8.4 8.5 8.6 Stationarity The Pair Correlation Function Local Properties Ergodicity A Spatial Ergodic Theorem Exercises 9 The Palm Distribution 9.1 9.2 9.3 9.4 9.5 Definition and Basic Properties The Mecke-Slivnyak Theorem Local Interpretation of Palm Distributions Voronoi Tessellations and the Inversion Formula Exercises 10 Extra Heads and Balanced Allocations 10.1 10.2 10.3 10.4 10.5 10.6 The Extra Head Problem The Point-Optimal Gale-Shapley Algorithm Existence of Balanced Allocations Allocations with Large Appetite The Modified Palm Distribution Exercises Mappings and Restrictions The Marking Theorem Thinnings Exercises 38 38 39 42 44 46 46 49 50 52 54 56 58 58 61 63 65 66 69 69 71 74 75 77 80 82 82 84 85 87 89 92 92 95 97 99 101 101
Contents xi 11 Stable Allocations 11.1 Stability 11.2 The Site-Optimal Gale-Shapley Allocation 11.3 Optimality of the Gale-Shapley Algorithms 11.4 Uniqueness of Stable Allocations 11.5 Moment Properties 11.6 Exercises 103 103 104 104 107 108 109 12 Poisson Integrals 12.1 The Wiener-Itô Integral 12.2 Higher Order Wiener-Itô Integrals 12.3 Poisson U-Statistics 12.4 Poisson Hyperplane Processes 12.5 Exercises 111 111 114 118 122 124 Random Measures and Cox Processes 13 13.1 Random Measures 13.2 Cox Processes 13.3 The Mecke Equation for Cox Processes 13.4 Cox Processes on Metric Spaces 13.5 Exercises 127 127 129 131 132 133 14 Permanentai Processes 14.1 Definition and Uniqueness 14.2 The Stationary Case 14.3 Moments of Gaussian Random Variables 14.4 Construction of Permanentai Processes 14.5 Janossy Measures of Permanentai Cox Processes 14.6 One-Dimensional Marginals of Permanentai Cox Processes 14.7 Exercises 136 136 138 139 141 145 147 151 15 Compound Poisson Processes 15.1 Definition and Basic Properties 15.2 Moments of Symmetric Compound Poisson Processes 15.3 Poisson Representation of Completely Random Measures 15.4 Compound Poisson Integrals 15.5 Exercises 153 153 157 158 161 163 16 The Boolean Model and the Gilbert Graph 16.1 Capacity Functional 16.2 Volume Fraction and Covering Property 16.3 Contact Distribution Functions 16.4 The Gilbert Graph 166 166 168 170 171
Contents xii 16.5 16.6 The Point Process of Isolated Nodes Exercises 176 177 17 The Boolean Model with General Grains 17.1 Capacity Functional 17.2 Spherical Contact Distribution Function and Covariance 17.3 Identifiability of Intensity and Grain Distribution 17.4 Exercises 179 179 182 183 185 Fock Space and Chaos Expansion 18 18.1 Difference Operators 18.2 Fock Space Representation 18.3 The Poincaré Inequality 18.4 Chaos Expansion 18.5 Exercises 187 187 189 193 194 195 Perturbation Analysis 19 19.1 A Perturbation Formula 19.2 Power Series Representation 19.3 Additive Functions of the Boolean Model 19.4 Surface Density of the Boolean Model 19.5 Mean Euler Characteristic of a Planar Boolean Model 19.6 Exercises 197 197 200 203 206 207 208 Covariance Identities 20 20.1 Mehler’s Formula 20.2 Two Covariance Identities 20.3 The Harris-FKG Inequality 20.4 Exercises 211 211 214 217 217 Normal Approximation 21 21.1 Stein’s Method 21.2 Normal Approximation via Difference Operators 21.3 Normal Approximation of Linear Functionals 21.4 Exercises 219 219 221 225 226 Normal Approximation in the Boolean Model 22 22.1 Normal Approximation of the Volume 22.2 Normal Approximation of Additive Functionals 22.3 Central Limit Theorems 22.4 Exercises 227 227 230 235 237
Contents xiii Appendix A Some Measure Theory A.l General Measure Theory A.2 Metric Spaces A.3 Hausdorff Measures and Additive Functionals A.4 Measures on the Real Half-Line A.5 Absolutely Continuous Functions 239 239 250 252 257 259 Appendix В Some Probability Theory B.l Fundamentals B.2 Mean Ergodic Theorem B.3 The Central Limit Theorem and Stein’s Equation B.4 Conditional Expectations B.5 Gaussian Random Fields 261 261 264 266 268 269 Appendix C Historical Notes 272 References Index 281 289
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| spelling | Last, Günter (DE-588)1168245125 aut Lectures on the Poisson process Günter Last (Karlsruhe Institute of Technology), Mathew Penrose (University of Bath) Cambridge Cambridge University Press 2018 © 2018 xx, 293 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Institute of Mathematical Statistics textbooks 7 Includes bibliographical references and index Poisson processes Stochastic processes Probabilities Poisson-Prozess (DE-588)4174971-6 gnd rswk-swf Poisson-Prozess (DE-588)4174971-6 s DE-604 Penrose, Mathew (DE-588)1200685938 aut Institute of Mathematical Statistics textbooks 7 (DE-604)BV036598560 7 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029963105&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4174971-6 |
| title | Lectures on the Poisson process |
| title_auth | Lectures on the Poisson process |
| title_exact_search | Lectures on the Poisson process |
| title_full | Lectures on the Poisson process Günter Last (Karlsruhe Institute of Technology), Mathew Penrose (University of Bath) |
| title_fullStr | Lectures on the Poisson process Günter Last (Karlsruhe Institute of Technology), Mathew Penrose (University of Bath) |
| title_full_unstemmed | Lectures on the Poisson process Günter Last (Karlsruhe Institute of Technology), Mathew Penrose (University of Bath) |
| title_short | Lectures on the Poisson process |
| title_sort | lectures on the poisson process |
| topic | Poisson processes Stochastic processes Probabilities Poisson-Prozess (DE-588)4174971-6 gnd |
| topic_facet | Poisson processes Stochastic processes Probabilities Poisson-Prozess |
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