In teaching an elementary course in stochastic processes it was noticed that many seemingly deep results in point processes are readily accessible by the device of representing them in terms of random gap lengths between points. The possibility of representing point processes in terms of sequences of random variables rather than probability measures makes them mathemati cally simpler than general stochastic processes. Point processes can be studied using only the tools of elementary probability, that is the joint distributions of finitely many random variables. Given the wide applicability of…mehr
In teaching an elementary course in stochastic processes it was noticed that many seemingly deep results in point processes are readily accessible by the device of representing them in terms of random gap lengths between points. The possibility of representing point processes in terms of sequences of random variables rather than probability measures makes them mathemati cally simpler than general stochastic processes. Point processes can be studied using only the tools of elementary probability, that is the joint distributions of finitely many random variables. Given the wide applicability of point process models and the difficulty of access by the measure-theoretic route, it was determined that the simpler representation is of sufficient expository im portance to deserve emphasis. The present book is the result: it is specialized and short and therefore is called a monograph. In its development the material has been taught to several classes with pleasing results. Students have apparently understood theorems which by other methods appear difficult and deep. A few of the results, particularly on reliability, safety assessment and clustering, are original applied research. An alternative title for this monograph might be 'Point processes: What they are and what they are good for.
1 Introduction.- 1.1 Arrivals in time.- 1.2 Reliability.- 1.3 Safety assessment.- 1.4 Random stress and strength.- Notes on the literature.- Problems.- 2 Point processes.- 2.1 The probabilistic context.- 2.2 Two methods of representation.- 2.3 Parameters of point processes.- 2.4 Transformation to a process with constant arrival rate.- 2.5 Time between arrivals.- Notes on the literature.- Problems.- 3 Homogeneous Poisson processes.- 3.1 Definition.- 3.2 Characterization.- 3.3 Time between arrivals for the hP process.- 3.4 Relations to the uniform distribution.- 3.5 A process with simultaneous arrivals.- Notes on the literature.- Problems.- 4 Application of point processes to a theory of safety assessment.- 4.1 The Reactor Safety Study.- 4.2 The annual probability of a reactor accident.- 4.3 A stochastic consequence model.- 4.4 A concept of rare events.- 4.5 Common mode failures.- 4.6 Conclusion.- Notes on the literature.- Problems.- 5 Renewal processes.- 5.1 Probabilistic theory.- 5.2 The renewal process cannot model equipment wearout.- Notes on the literature.- Problems.- 6 Poisson processes.- 6.1 The Poisson model.- 6.2 Characterization of regular Poisson processes.- 6.3 Time between arrivals for Poisson processes.- 6.4 Further observations on software error detection.- Notes on the literature.- Problems.- 7 Superimposed processes.- Notes on the literature.- Problems.- 8 Markov point processes.- 8.1 Theory.- 8.2 The Poisson process.- 8.3 Facilitation and hindrance.- Notes on the literature.- Problems.- 9 Applications of Markov point processes.- 9.1 Egg-laying dispersal of the bean weevil.- 9.2 Application of facilitation - hindrance to the spatial distribution of benthic invertebrates.- 9.3 The Luria-Delbrück model.- 9.4 Chance placement of balls in cells.- 9.5 Amodel for multiple vehicle automobile accidents.- 9.6 Engels' model.- Notes on the literature.- Problems.- 10 The order statistics process.- 10.1 The sampling of lifetimes.- 10.2 Derivation from the Poisson process.- 10.3 A Poisson model of equipment wearout.- Notes on the literature.- Problems.- 11 Competing risk theory.- 11.1 Markov chain model.- 11.2 Classical competing risks.- 11.3 Competing risk presentation of reactor safety studies.- 11.4 Delayed fatalities.- 11.5 Proportional hazard rates.- Notes on the literature.- Problems.- Further reading.- Appendix 1 Probability background.- A1.1 Probability distributions.- A1.2 Expectation.- A1.3 Transformation of variables.- A1.4 The distribution of order statistics.- A1.5 Conditional probability.- A1.6 Operational methods in probability.- A1.7 Convergence concepts and results in the theory of probability.- Notes on the literature.- Appendix 2 Technical topics.- A2.1 Existence of point process parameters.- A2.2 No simultaneous arrivals.- Solutions to a few of the problems.- References.- Author index.
1 Introduction.- 1.1 Arrivals in time.- 1.2 Reliability.- 1.3 Safety assessment.- 1.4 Random stress and strength.- Notes on the literature.- Problems.- 2 Point processes.- 2.1 The probabilistic context.- 2.2 Two methods of representation.- 2.3 Parameters of point processes.- 2.4 Transformation to a process with constant arrival rate.- 2.5 Time between arrivals.- Notes on the literature.- Problems.- 3 Homogeneous Poisson processes.- 3.1 Definition.- 3.2 Characterization.- 3.3 Time between arrivals for the hP process.- 3.4 Relations to the uniform distribution.- 3.5 A process with simultaneous arrivals.- Notes on the literature.- Problems.- 4 Application of point processes to a theory of safety assessment.- 4.1 The Reactor Safety Study.- 4.2 The annual probability of a reactor accident.- 4.3 A stochastic consequence model.- 4.4 A concept of rare events.- 4.5 Common mode failures.- 4.6 Conclusion.- Notes on the literature.- Problems.- 5 Renewal processes.- 5.1 Probabilistic theory.- 5.2 The renewal process cannot model equipment wearout.- Notes on the literature.- Problems.- 6 Poisson processes.- 6.1 The Poisson model.- 6.2 Characterization of regular Poisson processes.- 6.3 Time between arrivals for Poisson processes.- 6.4 Further observations on software error detection.- Notes on the literature.- Problems.- 7 Superimposed processes.- Notes on the literature.- Problems.- 8 Markov point processes.- 8.1 Theory.- 8.2 The Poisson process.- 8.3 Facilitation and hindrance.- Notes on the literature.- Problems.- 9 Applications of Markov point processes.- 9.1 Egg-laying dispersal of the bean weevil.- 9.2 Application of facilitation - hindrance to the spatial distribution of benthic invertebrates.- 9.3 The Luria-Delbrück model.- 9.4 Chance placement of balls in cells.- 9.5 Amodel for multiple vehicle automobile accidents.- 9.6 Engels' model.- Notes on the literature.- Problems.- 10 The order statistics process.- 10.1 The sampling of lifetimes.- 10.2 Derivation from the Poisson process.- 10.3 A Poisson model of equipment wearout.- Notes on the literature.- Problems.- 11 Competing risk theory.- 11.1 Markov chain model.- 11.2 Classical competing risks.- 11.3 Competing risk presentation of reactor safety studies.- 11.4 Delayed fatalities.- 11.5 Proportional hazard rates.- Notes on the literature.- Problems.- Further reading.- Appendix 1 Probability background.- A1.1 Probability distributions.- A1.2 Expectation.- A1.3 Transformation of variables.- A1.4 The distribution of order statistics.- A1.5 Conditional probability.- A1.6 Operational methods in probability.- A1.7 Convergence concepts and results in the theory of probability.- Notes on the literature.- Appendix 2 Technical topics.- A2.1 Existence of point process parameters.- A2.2 No simultaneous arrivals.- Solutions to a few of the problems.- References.- Author index.
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