Non-homogeneous Markov chains and systems: theory and applications
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| Format: | Buch |
| Sprache: | English |
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Boca Raton ; London ; New York
CRC Press
2023
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| Ausgabe: | First edition |
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xix, 450 Seiten Illustrationen |
| ISBN: | 9781138034525 9781032378046 |
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| 250 | |a First edition | ||
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| adam_text | Contents Preface xiii Acknowledgments xxi 1 FOUNDATIONS OF PROBABILITYTHEORY 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 Introductory notes Some set theory and topology 1.2.1 Set theory useful in probability 1.2.2 Interesting topological spaces Important family of sets in probability theory Measurable spaces Probability spaces 1.5.1 Infinite probability spaces Filtration Random variables Integration with respect to a probability measure Indicator functions The space L2 is a Hilbert space Independent σ-algebras and random variables Convergence of sequences of random variables The laws of large numbers and the central limittheorem Conditional distributions and conditional expectations Change of measure Existence and uniqueness of conditional expectations Properties of conditional expectation 2 A SMALL REVIEW OF MATRIXANALYSIS 2.1 2.2 Introductory notes Matrices 2.2.1 Similarity 2.3 The minimal polynomial of A 2.3.1 Invariant polynomials and elementary divisors 2.3.2 The Jordan canonical form 2.3.3 Remarks on the Jordan canonical form of a matrix 2.3.4 Construction of the transformation matrix of A to its Jordan form 2.4 The norm of a vector 2.5 The matrix norm 2.6 The Kronecker product of twomatrices 2.7 The Hadamard product of matrices 2.8 Canonical forms of a matrix 2.9 Generalized inverses 2.10 The Moore-Penrose generalized matrix 1 1 2 2 5 7 9 14 17 20 21 25 30 31 33 33 37 38 40 47 49 53 53 54 56 58 59 61 63 65 66 69 77 81 83 89 91 vii
viii CONTENTS 2.11 The Drazin inverse and the group inverse 2.12 The group inverse and Markov chains 2.13 Sensitivity of Markov chains 3 NON-HOMOGENEOUS MARKOV CHAINS; WEAK ERGODICITY 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 4.1 Strong ergodicity 4.2 Ergodicity and geometricstrong ergodicity 4.3 Criteria for strongergodicity for NHMC 4.4 Convergence in the Cesaro sense 4.4.1 Cyclic subclasses 4.4.2 The non-homogeneous case 4.5 Uniform strong ergodicity with the use of mean visit times 4.6 Strong ergodicity for general productsof matrices 4.7 Sets of matrices all products of which converge 4.8 A geometric approach to ergodic NHMC 4.9 Asymptotic behavior with arbitrary stochastic matrices 4.9.1 An illustrative example 5 THE NON-HOMOGENEOUS MARKOV SYSTEM 5.4 5.5 5.6 5.7 103 Introductory notes 103 Stochastic processes 103 Markov chain 105 3.3.1 A more advanced definition of a non-homogeneous Markov chain 106 The life and work of A.A. Markov 107 Probability distribution in the states of an NHMC 108 Examples 110 3.6.1 A more advanced result on the relation between a G֊non-homogeneous Markov chain and martingales 112 Weak and strong ergodicity 114 Structures for coefficients of ergodicity 115 Conditions for weak ergodicity for general products 122 3.9.1 A synopsis of the life and mathematical legacy of Wolfgang Doeblin 122 3.9.2 General products of stochastic matrices 123 3.9.3 Consequences of the above general theorems to weak ergodicity of inhomogeneous Markov chains 127 3.9.4 An application of backward products in non-homogeneous Markov chains 133 The dominant role of
the Dobrushin ergodicity coefficient 135 Transition probability matrices are in chronological order 141 Examples on the use of weak ergodicity theorems 145 4 NON-HOMOGENEOUS MARKOV CHAINS; STRONG ERGODICITY 5.1 5.2 5.3 92 98 99 Introductory notes The non-homogeneous Markov system in discrete time and space The expected and relative expected population structure 5.3.1 An expanding NHMS 5.3.2 A fluctuating NHMS without forced wastage A range of NHMS’s environment The ö-non-homogeneous Markov system Change of measure in ^-non-homogeueous Markov systems The space of random population structures as a Hilbert space 147 147 148 151 159 159 161 167 172 175 176 181 187 191 191 192 194 194 197 199 209 211 215
CONTENTS ix 5.8 Estimation of the transition probabilities of an NHMS 5.9 Research notes 218 219 6 ASYMPTOTIC BEHAVIOR OF A NON-HOMOGENEOUS MARKOV SYSTEM 221 6.1 Introductory notes 221 6.2 Asymptotic behavior of the expected population structure 221 6.3 The relative expected population structure 223 6.4 A contracting NHMS without forced wastage 227 6.5 NHMS with non-homogeneous Poisson recruitment 229 6.6 Asymptotic stability in NHMS’s 231 6.7 NHMS as a martingale with Poisson input 233 6.8 Research notes 236 7 ASYMPTOTIC VARIABILITY OF NHMS 7.1 Introductory notes 7.2 The V matrices and their properties 7.3 An illustrative probable application 7.4 Rate of convergence of the variability vector 7.5 Research notes 8 9 239 239 239 248 251 254 CYCLIC BEHAVIOR OF NON-HOMOGENEOUS MARKOV SYSTEMS 8.1 Introductory notes 8.2 Cyclic non-hoinogeneous Markov systems 8.3 An illustrative example from a manpower system 8.4 Rate of convergence of {μ (t)}^o ա an NHMSunder cyclic behavior 8.5 Research notes STOCHASTIC CONTROL IN NHMS’S 9.1 Introductory notes 9.2 Maintainability in NHMS by input control 9.3 Attainability in NHMS by input control 9.4 Asymptotically attainable structures 9.5 Periodicity of asymptotically attainable structures 9.6 An illustrative example 9.7 Control of asymptotic variability in NHMS 9.8 The NHMS in a stochastic environment 9.8.1 The expected population structure of the S-NHMS 9.8.2 Evaluating the expected transition probability matrix E[P(r)] 9.8.3 Expected value of a random number of Bernoulli trials with probability of success a random variable 9.8.4 The
expected population structure 9.9 Maintainability in a stochastic environment 9.10 Strategies for attaining a structure in an S-NHMS 9.11 An illustrative application 9.12 Research notes 255 255 255 260 262 262 263 263 263 266 268 271 285 286 289 290 290 292 292 293 296 298 298
X CONTENTS 10 LAWS OF LARGE NUMBERS FOR NON-HOMOGENEOUS MARKOV SYSTEMS 301 10.1 Introductory notes 301 10.2 Basic concepts and useful results 302 10.3 Laws of large numbers for an NHMS 303 10.4 An illustrative application 308 10.5 Laws of large numbers for a Cy-NHMS 310 10.6 An illustrative application 313 10.7 LLN for NHMS with arbitrary probability matrices 314 10.7.1 Cesaro convergence for Markov chains with arbitrary transition probability matrices 314 10.7.2 Cesaro convergence for an NHMS with arbitrary transition probability matrices for the inherent Markov chain 319 10.7.3 Law of large numbers for Cesaro sums of expected population structures 323 10.8 An illustrative example 326 11 THE S-NHMS IN CONTINUOUS TIME 11.1 Introductory notes 11.2 Asymptotic behavior of NHMP in continuous time 11.3 The S-NHMS in continuous time 11.4 The expected population structure of the S-NHMSC 11.5 The asymptotic behavior of the S-NHMSC 11.6 An illustrative example from manpower planning 11.7 Research notes 329 329 329 334 335 338 344 345 12 THE PERTURBED NON-HOMOGENEOUSMARKOV SYSTEM 12.1 Introductory notes 12.2 The group inverse and the asymptotic behavior 12.3 The perturbed non-homogeneous Markov system 12.4 The oscillation of the matrix Q in manpower systems 12.5 Asymptotic behavior of the P-NHMS 12.6 Sensitivity of the limiting distributions 12.7 Asymptotic variability of the P-NHMS 12.8 Research notes 347 347 347 350 351 351 355 357 360 13 NON-HOMOGENEOUS MARKOV SET SYSTEM 13.1 Introduction 13.2 Non-homogeneous Markov set system 13.3 The set of the expected relative population
structures 13.4 Asymptotic behavior of NHMSS 13.5 Properties of the limiting set Png^, (S, Ro) 13.6 An illustrative representative example 361 361 363 364 367 371 375 14 MARKOV SYSTEMS ON A GENERAL STATE SPACE 14.1 Introductory notes 14.2 A mental image for MSGS 14.3 The foundation of a MSGS 14.4 Asymptotic behavior or ergodicity of MSGS 14.5 Total variability from the invariant measure; a coupling theorem 14.6 Rate of convergence of MSGS 14.7 Asymptotic periodicity of a MSGS 14.8 Total variability from the invariant measures 381 381 382 382 388 392 393 397 403
CONTENTS xi 15 THE ^NON-HOMOGENEOUS MARKOV SYSTEM OF HIGH ORDER 15.1 Introductory notes 15.2 The ^-non-liomogeiieous Markov system of high order 15.3 The population structure of the Ç-NHMS 15.4 The inhomogeneous mixture transition distributionmodel 15.5 The asymptotic behavior of the inherent Markov chain 15.6 The asymptotic expected population structure 15.7 An illustrative example from manpower systems 15.8 Research notes 411 411 411 413 414 416 417 419 420 REFERENCES 421 INDEX 443
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| adam_txt |
Contents Preface xiii Acknowledgments xxi 1 FOUNDATIONS OF PROBABILITYTHEORY 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 Introductory notes Some set theory and topology 1.2.1 Set theory useful in probability 1.2.2 Interesting topological spaces Important family of sets in probability theory Measurable spaces Probability spaces 1.5.1 Infinite probability spaces Filtration Random variables Integration with respect to a probability measure Indicator functions The space L2 is a Hilbert space Independent σ-algebras and random variables Convergence of sequences of random variables The laws of large numbers and the central limittheorem Conditional distributions and conditional expectations Change of measure Existence and uniqueness of conditional expectations Properties of conditional expectation 2 A SMALL REVIEW OF MATRIXANALYSIS 2.1 2.2 Introductory notes Matrices 2.2.1 Similarity 2.3 The minimal polynomial of A 2.3.1 Invariant polynomials and elementary divisors 2.3.2 The Jordan canonical form 2.3.3 Remarks on the Jordan canonical form of a matrix 2.3.4 Construction of the transformation matrix of A to its Jordan form 2.4 The norm of a vector 2.5 The matrix norm 2.6 The Kronecker product of twomatrices 2.7 The Hadamard product of matrices 2.8 Canonical forms of a matrix 2.9 Generalized inverses 2.10 The Moore-Penrose generalized matrix 1 1 2 2 5 7 9 14 17 20 21 25 30 31 33 33 37 38 40 47 49 53 53 54 56 58 59 61 63 65 66 69 77 81 83 89 91 vii
viii CONTENTS 2.11 The Drazin inverse and the group inverse 2.12 The group inverse and Markov chains 2.13 Sensitivity of Markov chains 3 NON-HOMOGENEOUS MARKOV CHAINS; WEAK ERGODICITY 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 4.1 Strong ergodicity 4.2 Ergodicity and geometricstrong ergodicity 4.3 Criteria for strongergodicity for NHMC 4.4 Convergence in the Cesaro sense 4.4.1 Cyclic subclasses 4.4.2 The non-homogeneous case 4.5 Uniform strong ergodicity with the use of mean visit times 4.6 Strong ergodicity for general productsof matrices 4.7 Sets of matrices all products of which converge 4.8 A geometric approach to ergodic NHMC 4.9 Asymptotic behavior with arbitrary stochastic matrices 4.9.1 An illustrative example 5 THE NON-HOMOGENEOUS MARKOV SYSTEM 5.4 5.5 5.6 5.7 103 Introductory notes 103 Stochastic processes 103 Markov chain 105 3.3.1 A more advanced definition of a non-homogeneous Markov chain 106 The life and work of A.A. Markov 107 Probability distribution in the states of an NHMC 108 Examples 110 3.6.1 A more advanced result on the relation between a G֊non-homogeneous Markov chain and martingales 112 Weak and strong ergodicity 114 Structures for coefficients of ergodicity 115 Conditions for weak ergodicity for general products 122 3.9.1 A synopsis of the life and mathematical legacy of Wolfgang Doeblin 122 3.9.2 General products of stochastic matrices 123 3.9.3 Consequences of the above general theorems to weak ergodicity of inhomogeneous Markov chains 127 3.9.4 An application of backward products in non-homogeneous Markov chains 133 The dominant role of
the Dobrushin ergodicity coefficient 135 Transition probability matrices are in chronological order 141 Examples on the use of weak ergodicity theorems 145 4 NON-HOMOGENEOUS MARKOV CHAINS; STRONG ERGODICITY 5.1 5.2 5.3 92 98 99 Introductory notes The non-homogeneous Markov system in discrete time and space The expected and relative expected population structure 5.3.1 An expanding NHMS 5.3.2 A fluctuating NHMS without forced wastage A range of NHMS’s environment The ö-non-homogeneous Markov system Change of measure in ^-non-homogeueous Markov systems The space of random population structures as a Hilbert space 147 147 148 151 159 159 161 167 172 175 176 181 187 191 191 192 194 194 197 199 209 211 215
CONTENTS ix 5.8 Estimation of the transition probabilities of an NHMS 5.9 Research notes 218 219 6 ASYMPTOTIC BEHAVIOR OF A NON-HOMOGENEOUS MARKOV SYSTEM 221 6.1 Introductory notes 221 6.2 Asymptotic behavior of the expected population structure 221 6.3 The relative expected population structure 223 6.4 A contracting NHMS without forced wastage 227 6.5 NHMS with non-homogeneous Poisson recruitment 229 6.6 Asymptotic stability in NHMS’s 231 6.7 NHMS as a martingale with Poisson input 233 6.8 Research notes 236 7 ASYMPTOTIC VARIABILITY OF NHMS 7.1 Introductory notes 7.2 The V matrices and their properties 7.3 An illustrative probable application 7.4 Rate of convergence of the variability vector 7.5 Research notes 8 9 239 239 239 248 251 254 CYCLIC BEHAVIOR OF NON-HOMOGENEOUS MARKOV SYSTEMS 8.1 Introductory notes 8.2 Cyclic non-hoinogeneous Markov systems 8.3 An illustrative example from a manpower system 8.4 Rate of convergence of {μ (t)}^o ա an NHMSunder cyclic behavior 8.5 Research notes STOCHASTIC CONTROL IN NHMS’S 9.1 Introductory notes 9.2 Maintainability in NHMS by input control 9.3 Attainability in NHMS by input control 9.4 Asymptotically attainable structures 9.5 Periodicity of asymptotically attainable structures 9.6 An illustrative example 9.7 Control of asymptotic variability in NHMS 9.8 The NHMS in a stochastic environment 9.8.1 The expected population structure of the S-NHMS 9.8.2 Evaluating the expected transition probability matrix E[P(r)] 9.8.3 Expected value of a random number of Bernoulli trials with probability of success a random variable 9.8.4 The
expected population structure 9.9 Maintainability in a stochastic environment 9.10 Strategies for attaining a structure in an S-NHMS 9.11 An illustrative application 9.12 Research notes 255 255 255 260 262 262 263 263 263 266 268 271 285 286 289 290 290 292 292 293 296 298 298
X CONTENTS 10 LAWS OF LARGE NUMBERS FOR NON-HOMOGENEOUS MARKOV SYSTEMS 301 10.1 Introductory notes 301 10.2 Basic concepts and useful results 302 10.3 Laws of large numbers for an NHMS 303 10.4 An illustrative application 308 10.5 Laws of large numbers for a Cy-NHMS 310 10.6 An illustrative application 313 10.7 LLN for NHMS with arbitrary probability matrices 314 10.7.1 Cesaro convergence for Markov chains with arbitrary transition probability matrices 314 10.7.2 Cesaro convergence for an NHMS with arbitrary transition probability matrices for the inherent Markov chain 319 10.7.3 Law of large numbers for Cesaro sums of expected population structures 323 10.8 An illustrative example 326 11 THE S-NHMS IN CONTINUOUS TIME 11.1 Introductory notes 11.2 Asymptotic behavior of NHMP in continuous time 11.3 The S-NHMS in continuous time 11.4 The expected population structure of the S-NHMSC 11.5 The asymptotic behavior of the S-NHMSC 11.6 An illustrative example from manpower planning 11.7 Research notes 329 329 329 334 335 338 344 345 12 THE PERTURBED NON-HOMOGENEOUSMARKOV SYSTEM 12.1 Introductory notes 12.2 The group inverse and the asymptotic behavior 12.3 The perturbed non-homogeneous Markov system 12.4 The oscillation of the matrix Q in manpower systems 12.5 Asymptotic behavior of the P-NHMS 12.6 Sensitivity of the limiting distributions 12.7 Asymptotic variability of the P-NHMS 12.8 Research notes 347 347 347 350 351 351 355 357 360 13 NON-HOMOGENEOUS MARKOV SET SYSTEM 13.1 Introduction 13.2 Non-homogeneous Markov set system 13.3 The set of the expected relative population
structures 13.4 Asymptotic behavior of NHMSS 13.5 Properties of the limiting set Png^, (S, Ro) 13.6 An illustrative representative example 361 361 363 364 367 371 375 14 MARKOV SYSTEMS ON A GENERAL STATE SPACE 14.1 Introductory notes 14.2 A mental image for MSGS 14.3 The foundation of a MSGS 14.4 Asymptotic behavior or ergodicity of MSGS 14.5 Total variability from the invariant measure; a coupling theorem 14.6 Rate of convergence of MSGS 14.7 Asymptotic periodicity of a MSGS 14.8 Total variability from the invariant measures 381 381 382 382 388 392 393 397 403
CONTENTS xi 15 THE ^NON-HOMOGENEOUS MARKOV SYSTEM OF HIGH ORDER 15.1 Introductory notes 15.2 The ^-non-liomogeiieous Markov system of high order 15.3 The population structure of the Ç-NHMS 15.4 The inhomogeneous mixture transition distributionmodel 15.5 The asymptotic behavior of the inherent Markov chain 15.6 The asymptotic expected population structure 15.7 An illustrative example from manpower systems 15.8 Research notes 411 411 411 413 414 416 417 419 420 REFERENCES 421 INDEX 443 |
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| spelling | Vassiliou, Panos C. G. Verfasser (DE-588)142769819 aut Non-homogeneous Markov chains and systems theory and applications P.-C.G. Vassiliou, University College London, UK First edition Boca Raton ; London ; New York CRC Press 2023 © 2023 xix, 450 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Markov-Kette (DE-588)4037612-6 gnd rswk-swf Markov-System (DE-588)4427910-3 gnd rswk-swf Markov-Kette (DE-588)4037612-6 s Markov-System (DE-588)4427910-3 s Markov-Prozess (DE-588)4134948-9 s DE-604 Erscheint auch als Online-Ausgabe 978-1-315-26998-6 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=034011014&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vassiliou, Panos C. G. Non-homogeneous Markov chains and systems theory and applications Markov-Prozess (DE-588)4134948-9 gnd Markov-Kette (DE-588)4037612-6 gnd Markov-System (DE-588)4427910-3 gnd |
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| title | Non-homogeneous Markov chains and systems theory and applications |
| title_auth | Non-homogeneous Markov chains and systems theory and applications |
| title_exact_search | Non-homogeneous Markov chains and systems theory and applications |
| title_exact_search_txtP | Non-homogeneous Markov chains and systems theory and applications |
| title_full | Non-homogeneous Markov chains and systems theory and applications P.-C.G. Vassiliou, University College London, UK |
| title_fullStr | Non-homogeneous Markov chains and systems theory and applications P.-C.G. Vassiliou, University College London, UK |
| title_full_unstemmed | Non-homogeneous Markov chains and systems theory and applications P.-C.G. Vassiliou, University College London, UK |
| title_short | Non-homogeneous Markov chains and systems |
| title_sort | non homogeneous markov chains and systems theory and applications |
| title_sub | theory and applications |
| topic | Markov-Prozess (DE-588)4134948-9 gnd Markov-Kette (DE-588)4037612-6 gnd Markov-System (DE-588)4427910-3 gnd |
| topic_facet | Markov-Prozess Markov-Kette Markov-System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034011014&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT vassilioupanoscg nonhomogeneousmarkovchainsandsystemstheoryandapplications |