Adventures in Stochastic Processes

This textbook provides easy access to stochastic processes for students of applied science at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is very accessible to beginners but flexible enough to serve those who come to the course with strong backgrounds. Adventures in Stochastic Processes contains many examples, exercises, and applications of a practical and serious nature. Underlying principles of complex problems and computations are cleanly and quickly delineated through lively and rich vignettes of a personalized group of characters inhabiting our random world. Each chapter is fully supplemented with exercises. Highly Acclaimed Textbook! This text has been adopted at: Harvard University, Cambridge, MA Georgia Institute of Technology, Atlanta Cornell University, Ithaca, NY Bowling Green State University, Bowling Green, OH Elmhurst College, Elmhurst, IL University of North Carolina, Chapel Hill Kansas State University, Manhattan Iowa State University, Ames Texas Tech University, Lubbock McGill University, Montreal, Canada University of Toronto, Canada Ch. 1. Preliminaries: Discrete Index Sets And/or Discrete State Spaces. 1.1. Non-negative Integer Valued Random Variables. 1.2. Convolution. 1.3. Generating Functions. 1.4. The Simple Branching Process. 1.5. Limit Distributions And The Continuity Theorem. 1.6. The Simple Random Walk. 1.7. The Distribution Of A Process. 1.8. Stopping Times -- Ch. 2. Markov Chains. 2.1. Construction And First Properties. 2.2. Examples. 2.3. Higher Order Transition Probabilities. 2.4. Decomposition Of The State Space. 2.5. The Dissection Principle. 2.6. Transience And Recurrence. 2.7. Periodicity. 2.8. Solidarity Properties. 2.9. Examples. 2.10. Canonical Decomposition. 2.11. Absorption Probabilities. 2.12. Invariant Measures And Stationary Distributions. 2.13. Limit Distributions. 2.14. Computation Of The Stationary Distribution. 2.15. Classification Techniques -- Ch. 3. Renewal Theory. 3.1. Basics. 3.2. Analytic Interlude. 3.3. Counting Renewals. 3.4. Renewal Reward Processes. 3.5. The Renewal Equation. 3.6. The Poisson Process As A Renewal Process. 3.7. Informal Discussion Of Renewal Limit Theorems; Regenerative Processes. 3.8. Discrete Renewal Theory. 3.9. Stationary Renewal Processes. 3.10. Blackwell And Key Renewal Theorems. 3.11. Improper Renewal Equations. 3.12. More Regenerative Processes -- Ch. 4. Point Processes. 4.1. Basics. 4.2. The Poisson Process. 4.3. Transforming Poisson Processes. 4.4. More Transformation Theory; Marking And Thinning. 4.5. The Order Statistic Property. 4.6. Variants Of The Poisson Process. 4.7. Technical Basics. 4.8. More On The Poisson Process. 4.9. A General Construction Of The Poisson Process; A Simple Derivation Of The Order Statistic Property. 4.10. More Transformation Theory; Location Dependent Thinning. 4.11. Records -- Ch. 5. Continuous Time Markov Chains. 5.1. Definitions And Construction. 5.2. Stability And Explosions. 5.3. Dissection. 5.4. The Backward Equation And The Generator Matrix. 5.5. Stationary And Limiting Distributions. 5.6. Laplace Transform Methods. 5.7. Calculations And Examples. 5.8. Time Dependent Solutions. 5.9. Reversibility. 5.10. Uniformizability. 5.11. The Linear Birth Process As A Point Process -- Ch. 6. Brownian Motion. 6.1. Introduction. 6.2. Preliminaries. 6.3. Construction Of Brownian Motion. 6.4. Simple Properties Of Standard Brownian Motion. 6.5. The Reflection Principle And The Distribution Of The Maximum. 6.6. The Strong Independent Increment Property And Reflection. 6.7. Escape From A Strip. 6.8. Brownian Motion With Drift. 6.9. Heavy Traffic Approximations In Queueing Theory. 6.10. The Brownian Bridge And The Kolmogorov-smirnov Statistic. 6.11. Path Properties. 6.12. Quadratic Variation. 6.13. Khintchine's Law Of The Iterated Logarithm For Brownian Motion -- Ch. 7. The General Random Walk. 7.1. Stopping Times. 7.2. Global Properties. 7.3. Prelude To Wiener-hopf: Probabilistic Interpretations Of Transforms. 7.4. Dual Pairs Of Stopping Times. 7.5. Wiener-hopf Decompositions. 7.6. Consequences Of The Wiener-hopf Factorization. 7.7. The Maximum Of A Random Walk. 7.8. Random Walks And The G/g/1 Queue. Sidney Resnick. Includes Bibliographical References (p. [613]-616) And Index. Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. In a lively and imaginative presentation, studded with examples, exercises, and applications, and supported by inclusion of computational procedures, the author has created a textbook that provides easy access to this fundamental topic for many students of applied sciences at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is accessible to beginners but flexible enough to serve as well those who come to the course with strong backgrounds. The prerequisite background for reading the book is a graduate level pre-measure theoretic probability course. No knowledge of measure theory is presumed and advanced notions of conditioning are scrupulously avoided until the later chapters of the book. The book can be used for either a one or two semester course as given in departments of mathematics, statistics, operation research, business and management, or a number of engineering departments. Its approach to exercises and applications is practical and serious. Some underlying principles of complex problems and computations are cleanly and quickly delineated through rich vignettes of whimsically imagined Happy Harry and his Optima Street gang’s adventures in a world whose randomness is a never-ending source of both wonder and scientific insight. The tools of applied probability---discrete spaces, Markov chains, renewal theory, point processes, branching processes, random walks, Brownian motion---are presented to the reader in illuminating discussion. Applications include such topics as queuing, storage, risk analysis, genetics, inventory, choice, economics, sociology, and other. Because of the conviction that analysts who build models should know how to build them for each class of process studied, the author has included such constructions. Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. This text offers easy access to this fundamental topic for many students of applied sciences at many levels. It includes examples, exercises, applications, and computational procedures. It is uniquely useful for beginners and non-beginners in the field. No knowledge of measure theory is presumed. Pradžia......Page 2 Turinys......Page 3 Įžanga......Page 5 1 skyrius. Pagalbinis......Page 7 2 skyrius. Markovo grandinės......Page 37 3 skyrius. Atstatymo procesai......Page 94 4 skyrius. Taškiniai procesai......Page 157 5 skyrius. Tolydaus laiko Markovo grandinės......Page 190 6 skyrius. Brauno judėjimas......Page 248 7 skyrius. Atsitiktinis klaidžiojimas......Page 286 Bibliografija......Page 313