随机过程探究

preface
chapter 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
　　1.3.1. differentiation of generating functions
　　1.3.2. generating functions and moments
　　1.3.3. generating functions and convolution
　　1.3.4. generating functions, compounding and random sums
　1.4. the simple branching process
　1.5. limit distributions and the continuity theorem
　　1.5.1. the law of rare events
　1.6. the simple random walk
　1.7. the distribution of a process*
　1.8. stopping times*
　　1.8.1. wald's identity
　　1.8.2. splitting an lid sequence at a stopping time
　exercises for chapter 1
chapter 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
　　2.12.1. time averages
　2.13. limit distributions
　　2.13.1 more on null recurrence and transience*
　2.14. computation of the stationary distribution
　2.15. classification techniques
　exercises for chapter 2
chapter 3. renewal theory
　3.1. basics
　3.2. analytic interlude
　　3.2.1. integration
　　3.2.2. convolution
　　3.2.3. laplace transforms
　3.3. counting renewals
　3.4. renewal reward processes
　3.5. the renewal equation
　　3.5.1. risk processes*
　3.6. the poisson process as a renewal process
　3.7. informal discussion of renewal limit theorems;regenerative processes
　　3.7.1 an informal discussion of regenerative processes
　3.8. discrete renewal theory
　3.9. stationary renewal processes*
　3.10. blackwell and key renewal theorems*
　　3.10.1. direct riemann integrability*
　　3.10.2. equivalent forms of the renewal theorems*
　　3.10.3. proof of the renewal theorem*
　3.11. improper renewal equations
　3.12. more regenerative processes*
　　3.12.1. definitions and examples*
　　3.12.2. the renewal equation and smith's theorem*
　　3.12.3. queueing examples
　exercises for chapter 3
chapter 4.point processes
　4.1. basics
　4.2. the poisson process
　4.3. transforming poisson processes
　　4.3.1. max-stable and stable random variables*
　4.4. more transformation theory; marking and thinning
　4.5. the order statistic property
　4.6. variants of the poiason process
　4.7. wec. hnlcal basics*
　　4.7.1. the laplace functional*
　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*
　exercises for chapter 4
chapter 5.continuous time markov chains
　5.1. definitions and construction
　5.2. stability and explosions
　　5.2.1. the markov property*
　5.3. dissection
　　5.3.1. more detail on dissection*
　5.4. the backward equation and the generator matrix
　5.5. stationary and limiting distributions
　　5.5.1. more on invariant measures*
　5.6. laplace transform methods
　5.7. calculations and examples
　　5.7.1. queueing networks
　5.8. time dependent solutions*
　5.9. reversibility
　5.10. uniformizability
　5.11. the linear birth process as a point process
　exercises for chapter 5
chapter 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*
　exercises for chapter 6
chapter 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
　　7.8.1. exponential right tail
　　7.8.2. application to g/m/1 queueing model,
　　7.8.3. exponential left tail
　　7.8.4. the m/g/1 queue
　　7.8.5. queue lengths
references
index