Author | Schinazi, Rinaldo B. author |
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Title | Classical and Spatial Stochastic Processes [electronic resource] / by Rinaldo B. Schinazi |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1582-0 |
Descript | XI, 179 p. online resource |
I Discrete Time Markov Chains -- I.1 Three fundamental examples -- I.2 Classification of states -- I.3 Finite Markov chains -- I.4 Birth and death chains -- I.5 An example of coupling -- I.6 Time to get ruined -- I.7 Absorption probabilities for martingales -- I.8 Random walks -- I.9 The Bienaymรฉ-Galton-Watson branching process -- I.10 Proof of Theorem I.2.1 -- I.11 Proof of Theorem I.2.2 -- I.12 Proof of Theorem I.9.1 -- II Stationary Distributions of a Markov Chain -- II. 1 Existence of stationary distributions -- II.2 Reversible measures -- II.3 Convergence to a stationary distribution -- II.4 The finite case -- II.5 Proof of Proposition II. 1.2 -- II.6 Proof of Proposition II. 1.3 -- II.7 Proofs of Theorems II.3.1 and II.4.2 -- III Continuous Time Birth and Death Markov Chains -- III.1 The exponential distribution -- III.2 Construction and first properties -- III.3 Limiting probabilities -- III.4 Classification of states -- III.5 The Poisson process -- III.6 Passage times -- III.7 A queue which is not Markovian -- III.8 Proof of Theorem III.3.1 -- III.9 Proof of Theorem III.5.1 -- III.10 Proof of Theorem III.5.2 -- IV Percolation -- IV.1 Percolation on Zd -- IV.2 Further properties of percolation on Zd -- IV.3 Percolation on a tree and two critical exponents -- V A Cellular Automaton -- V.1 The model -- V.2 A renormalization argument -- VI Continuous Time Branching Random Walk -- VI.1 A continuous time Bienayme-Galton-Watson process -- VI.2 A continuous time branching random walk -- VI.3 The first phase transition is continuous -- VI.4 The second phase transition is discontinuous -- VI.5 Proof of Theorem VI.2.1 -- VII The Contact Process on a Homogeneous Tree -- VII.1 The two phase transitions -- VII.2 Characterization of the first phase transition -- VII.3 The graphical construction -- VII.4 Proofs -- VII.5 Open problems -- Appendix: Some Facts About Probabilities on Countable Spaces -- 1 Probability space -- 2 Independence -- 3 Discrete random variables -- References