while this is a book about harry and hisadventurous life, it is primarily a serious text about stochasticprocesses. it features the basic stochas-tic processes that arenecessary ingredients for building models of a wide variety ofphenomena exhibiting time varying randomness.
the book is intended as a first year graduate text for coursesusually called stochastic processes (perhaps amended by the words"applied" or "introduction to ... ") or applied probability, orsometimes stochastic modelling. it is meant to be very accessibleto beginners, and at the same time, to serve those who come to thecourse with strong backgrounds. this flexiblity also permits theinstructor to push the sophistication level up or down. for thenovice, discussions and motivation are given carefully and in greatdetail. in some sections beginners are advised to skip certaindevel-opments, while in others, they can read the words and skipthe symbols in order to get the content without more technicaldetail than they are ready to assimilate. in fact, with thenumerous readings and variety of prob-lems, it is easy to carve apath so that the book challenges more advanced students, butremains instructive and manageable for beginners. some sections arestarred and come with a warning that they contain material which ismore mathematically demanding. several discussions have beenmodularized to facilitate flexible adaptation to the needs ofstudents with differing backgrounds. the text makes crystal cleardistinctions between the following: proofs, partial proofs,motivations, plausibility arguments and good old fashionedhand-waving.
preface
chapter 1. preliminaries: discrete index sets and/or discrete statespaces
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*
preface
chapter 1. preliminaries: discrete index sets and/or discrete statespaces
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;regenerativeprocesses
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 simplederivation of the order statistic property*
4.10. more transformation theory; location dependentthinning*
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 themaximum
6.6. the strong independent increment property andreflection*
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-smirnovstatistic
6.11. path properties*
6.12. quadratic variation
6.13. khintchine's law of the iterated logarithm for brownianmotion*
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 oftransforms
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
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