Dependence with complete connections and its applications:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1990
|
| Edition: | 1. publ. |
| Series: | Cambridge tracts in mathematics
96 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 304 S. |
| ISBN: | 0521333318 |
Staff View
MARC
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| 245 | 1 | 0 | |a Dependence with complete connections and its applications |c Marius Iosifescu ; Serban Grigorescu |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1990 | |
| 300 | |a XIV, 304 S. | ||
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Record in the Search Index
| _version_ | 1804118293765685248 |
|---|---|
| adam_text | Contents
Preface xi
Abbreviations and notation xiii
Introduction 1
1 Fundamental notions 5
1.1 The concept of a random system with complete connections 5
1.1.1 The homogeneous case 5
1.1.2 The existence theorem 6
1.1.3 The non-homogeneous case 10
1.1.4 Generalized systems 12
1.2 Examples 15
1.3 Classification problems 31
1.3.1 A communication relation 31
1.3.2 Random systems with complete connections of type (B) 34
Problems 36
2 Ergodidty 38
2.1 Implications of ergodicity 38
2.1.1 Preliminaries 38
2.1.2 A special ergodic theorem 40
2.1.3 A fundamental theorem 43
2.2 The homogeneous case 46
2.2.1 Preliminaries 46
2.2.2 Condition FLS(^o.v) 47
2.2.3 Necessary and sufficient conditions for uniform ergodicity 51
2.2.4 A sufficient condition for ergodicity 54
2.3 The non-homogeneous case 55
2.3.1 Weak-ergodicity 55
2.3.2 Strong-ergodicity 58
2.4 An application to the associated Markov chain 60
Problems 64
3 The associated Markov chain 68
3.1 Associated operators 68
viii Contents
3.1.1 Preliminaries 68
3.1.2 Doeblin s condition and (quasi-) compact operators 72
3.1.3 Doeblin-Fortet operators 76
3.1.4 Markov operators on C{W) 86
3.2 Compact Markov chains 93
3.3 Continuous Markov chains 99
3.3.1 Preliminaries 99
3.3.2 Ergodic kernels and transience 101
3.3.3 The case of one ergodic kernel 105
3.3.4 The general case (several ergodic kernels) 110
3.3.5 Subergodic decomposition, ergodicity, regularity 114
3.3.6 The structure of the tail ff-algebra 118
3.4 Applications to ergodicity 122
Problems 126
4 Asymptotic behaviour 132
4.1 Limit theorems for the infinite order chain 132
4.1.1 Preliminaries 132
4.1.2 The law of large numbers 133
4.1.3 The functional central limit theorem 134
4.1.4 The central limit theorem with remainder 141
4.1.5 The law of the iterated logarithm 143
4.1.6 Some non-parametric statistics 144
4.2 Limit theorems for the associated Markov chain 146
4.2.1 Preliminaries 146
4.2.2 The law of large numbers 147
4.2.3 The central limit theorem with remainder 148
4.2.4 An invariance principle for the central limit theorem
and the law of the iterated logarithm 155
Problems 156
5 Some special systems 160
5.1 OM chains 160
5.1.1 Preliminaries 160
5.1.2 Ergodicity 162
5.1.3 Alternate linear OM chains 169
5.1.4 The case of an arbitrary state space 173
5.2 The continued fraction expansion 174
5.2.1 Preliminaries 174
5.2.2 Gauss problem: Levy s approach 179
5.2.3 Gauss problem: Kuzmin s approach 183
5.2.4 Limit theorems 186
5.2.5 Generalizations 189
5.3 Piecewise monotonic transformations 190
Contents ix
5.3.1 Preliminaries 190
5.3.2 The invariant probability: existence and uniqueness 195
5.3.3 The Frobenius-Perron operator 197
5.3.4 The invariant probability: differential properties of its density 205
5.3.5 Limit theorems 209
5.3.6 A few final remarks 214
5.4/-expansions 215
5.4.1 Preliminaries 215
5.4.2 Gauss problem and mixing properties 218
5.4.3 A conjecture 222
5.4.4 The D-ary expansion 222
5.5 Strict-sense infinite-order chains 225
5.5.1 Preliminaries 225
5.5.2 An existence theorem 226
5.5.3 The case of a countable state space 228
5.5.4 The case of a finite state space 232
Problems 234
Appendix 1 Spaces, measures and functions 241
Appendix 2 Notions of functional analysis 256
Appendix 3 Mixing and Markovian dependence 268
Notes and comments 277
Reference 284
Index 303
|
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| author | Iosifescu, Marius 1936-2020 Grigorescu, Serban |
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| dewey-full | 519.2 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
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| discipline | Mathematik |
| edition | 1. publ. |
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| indexdate | 2024-07-09T16:08:00Z |
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| isbn | 0521333318 |
| language | English |
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| physical | XIV, 304 S. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge tracts in mathematics |
| series2 | Cambridge tracts in mathematics |
| spelling | Iosifescu, Marius 1936-2020 Verfasser (DE-588)107476444 aut Dependence with complete connections and its applications Marius Iosifescu ; Serban Grigorescu 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1990 XIV, 304 S. txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 96 Connections (Mathématiques) Connexions (mathématiques) ram Markov, Processus de ram Processus de Markov Systèmes stochastiques Systèmes stochastiques ram Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s DE-604 Grigorescu, Serban Verfasser aut Cambridge tracts in mathematics 96 (DE-604)BV000000001 96 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002540922&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Iosifescu, Marius 1936-2020 Grigorescu, Serban Dependence with complete connections and its applications Cambridge tracts in mathematics Connections (Mathématiques) Connexions (mathématiques) ram Markov, Processus de ram Processus de Markov Systèmes stochastiques Systèmes stochastiques ram Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4057630-9 |
| title | Dependence with complete connections and its applications |
| title_auth | Dependence with complete connections and its applications |
| title_exact_search | Dependence with complete connections and its applications |
| title_full | Dependence with complete connections and its applications Marius Iosifescu ; Serban Grigorescu |
| title_fullStr | Dependence with complete connections and its applications Marius Iosifescu ; Serban Grigorescu |
| title_full_unstemmed | Dependence with complete connections and its applications Marius Iosifescu ; Serban Grigorescu |
| title_short | Dependence with complete connections and its applications |
| title_sort | dependence with complete connections and its applications |
| topic | Connections (Mathématiques) Connexions (mathématiques) ram Markov, Processus de ram Processus de Markov Systèmes stochastiques Systèmes stochastiques ram Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Connections (Mathématiques) Connexions (mathématiques) Markov, Processus de Processus de Markov Systèmes stochastiques Stochastischer Prozess |
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