随机过程探究(英文版)

　　 while this is a book about harry and his adventurous life, it is primarily a serious text about stochastic processes. it features the basic stochas-tic processes that are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness.
　　 the book is intended as a first year graduate text for courses usually called stochastic processes (perhaps amended by the words "applied" or "introduction to ... ") or applied probability, or sometimes stochastic modelling. it is meant to be very accessible to beginners, and at the same time, to serve those who come to the course with strong backgrounds. this flexiblity also permits the instructor to push the sophistication level up or down. for the novice, discussions and motivation are given carefully and in great detail. in some sections beginners are advised to skip certain devel-opments, while in others, they can read the words and skip the symbols in order to get the content without more technical detail than they are ready to assimilate. in fact, with the numerous readings and variety of prob-lems, it is easy to carve a path so that the book challenges more advanced students, but remains instructive and manageable for beginners. some sections are starred and come with a warning that they contain material which is more mathematically demanding. several discussions have been modularized to facilitate flexible adaptation to the needs of students with differing backgrounds. the text makes crystal clear distinctions between the following: proofs, partial proofs, motivations, plausibility arguments and good old fashioned hand-waving.

《随机过程探究(英文版)》
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

　　 While this is a book about Harry and his adventurous life, it is primarily a serious text about stochastic processes. It features the basic stochas-tic processes that are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness.
　　 The book is intended as a first year graduate text for courses usually called Stochastic Processes (perhaps amended by the words "Applied" or "Introduction to ... ") or Applied Probability, or sometimes Stochastic Modelling. It is meant to be very accessible to beginners, and at the same time, to serve those who come to the course with strong backgrounds. This flexiblity also permits the instructor to push the sophistication level up or down. For the novice, discussions and motivation are given carefully and in great detail. In some sections beginners are advised to skip certain devel-opments, while in others, they can read the words and skip the symbols in order to get the content without more technical detail than they are ready to assimilate. In fact, with the numerous readings and variety of prob-lems, it is easy to carve a path so that the book challenges more advanced students, but remains instructive and manageable for beginners. Some sections are starred and come with a warning that they contain material which is more mathematically demanding. Several discussions have been modularized to facilitate flexible adaptation to the needs of students with differing backgrounds. The text makes crystal clear distinctions between the following: proofs, partial proofs, motivations, plausibility arguments and good old fashioned hand-waving.
　　 Where did Harry, Zeke and the rest of the gang come from? Courses in Stochastic Processes tend to contain overstuffed curricula. It is, therefore,useful to have quick illustrations of how the theory leads to techniques for calculating numbers. With the Harry vignettes, the student can get in and out of numerical illustrations quickly. Of course, the vignettes are not meant to replace often stimulating but time consuming real applications.A variety of examples with applied appeal are sprinkled throughout the exposition and exercises. Our students are quite fond of Harry and enjoy psychoanalyzing him, debating whether he is "a polyester sort of guy"or the "jeans and running shoes type." They seem to have no trouble discerning the didactic intent of the Harry stories and accept the need for some easy numerical problems before graduating to more serious ones.Student culture has become so ubiquitous that foreign students who are not native English speakers can quickly get into the swing. I think Harry is a useful and entertaining guy but if you find that you loathe him, he is easy to avoid in the text.
　　 Where did theiy come.v
　　 I can't say.
　　 But I bet they have come a long long walt. 1 To the instructor: The discipline imposed during the writing was that the first six chapters should not use advanced notions of conditioning which involve relatively sophisticated ideas of integration. Only the elementary definition is used: P(A[B) = P(A 0 B)/P(B). Instead of conditioning arguments we find independence where we need it and apply some form of the product rule: P(A 0 B) = P(A)P(B) if A and B are independent.This maintains rigor and keeps the sophistication level down.
　　 No knowledge of measure theory is assumed but it is assumed that the student has already digested a good graduate level pre-measure theoretic probability course. A bit of measure theory is discussed here and there in starred portions of the text. In most cases it is simple and intuitive but if it scares you, skip it and you will not be disadvantaged as you journey through the book. If, however, you know some measure theory,you will understand things in more depth. There is a sprinkling of refer-ences throughout the book to Fubini's theorem, the monotone convergence theorem and the dominated convergence theorem. These are used to jus-tify the interchange of operations such as summation and integration. A relatively unsophisticated student would not and should not worry about justifications for these interchanges of operations; these three theorems should merely remind such students that somebody knows how to cheek the correctness of these interchanges.
　　 Analysts who build models are supposed to know how to build mod-els. So for each class of process studied, a construction of that process is included. Independent, identically distributed sequences are usually as-sumed as primitives in the constructions. Once a concrete version of the process is at hand, many properties are fairly transparent. Another benefit is that if you know how to construct a stochastic process, you know how to simulate the process. While no specific discussion of simulation is in-cluded here, I have tried to avoid pretending the computer does not exist.For instance, in the Markov chain chapters, formulas are frequently put in matrix form to make them suitable for solution by machine rather than by hand. Packages such as Minitab, Mathematica, Gauss, Matlab, etc., have been used successfully as valuable aids in the solution of problems but local availability of computing resources and the rapidly changing world of hard-ware and software make specific suggestions unwise. Ask your local guru for suggestions. You need to manipulate some matrices, and find roots of polynomials; but nothing too fancy. If you have access to a package that does symbolic calculations, so much the better. A companion disk to this book is being prepared by Douglas McBeth which will allow easy solutions to many numerical problems.
　　 There is much more material here than can be covered in one semester.Some selection according to the needs of the students is required. Here is the core of the material: Chapter 1: 1.1-1.6. Skip the proof of the continuity theorem in 1.5 if necessary but mention Wald's identity. Some instructors may prefer to skip Chapter 1 anti return later to these topics,as needed. If you are tempted by this strategy, keep in mind that Chapter 1 discusses the interesting and basic random walk and branching processes and that facility with transforms is worthwhile. Chapter 2: 2.1-2.12,2.12.1.In Section 2.13, a skilled lecturer is advised to skip most of the proof of Theorem 2.13.2, explain coupling in 15 minutes, and let it go at that. This is one place where hand-waving really conveys something. The material from Section 2.13.1 should be left to the curious. If time permits, try to cover Sections 2.14 and 2.15 but you will have to move at a brisk pace.Chapter 3: In renewal theory stick to basics. After all the discrete state space theory in Chapters 1 and 2, the switch to the continuous state space world leaves many students uneasy. The core is Sections 3.1-3.5, 3.6, 3.7,and 3.7.1. Sections 3.8 and 3.12.3 are accessible ff there is time but 3.9-3.12.2 are only for supplemental reading by advanced students. Chapter 4: The jeweis are in Sections 4.1 to 4.7. You can skip 4.3.1. Ifyou have a group that can cope with a bit more sophistication, try 4.7.1, 4.8 and 4.9. Once you come to know and love the Laplace functional, the rest is incredibly easy and short. Chapter 5: The basics are 5.1-5.8. If you are pressed for time, skip possibly 5.6 and 5.8; beginners may avoid 5.2.1, 5.3.1 and 5.5.1.Section 5.7.1 is on queueing networks and is a significant application of standard techniques, so try to reserve some time for it. Section 5.9 is nice if there is time. Despite its beauty, leave 5.11 for supplemental reading by advanced students. Chapter 6: Stick to some easy path properties,strong independent increments, reflection, and some explicit calculations.I recommend 6.1, 6.2, 6.4, 6.5, 6.6, 6.7, and 6.8. For beginners, a quick survey of 6.11-6.13 may be adequate. If there is time and strong interest in queueing, try 6.9. If there is strong interest in statistics, try 6.10. I like Chapter 7, but it is unlikely it can be covered in a first course. Parts of it require advanced material.
　　 In the course of teaching, I have collected problems which have been inserted into the examples and problem sections; there should be a good supply. These illustrate a variety of applied contexts where the skills mas-tered in the chapter can be used. Queueing theory is a frequent context for many exercises. Many problems emphasize calculating numbers which seems to be a skill most students need these days, especially considering the wide clientele who enroll for courses in stochastic processes. There is a big payoff for the student who will spend serious time working out the problems. Failure to do do will relegate the novice reader to the status of voyeur.
　　 Some acknowledgements and thank you's: The staff at Birkh~user has been very supportive, efficient and colleagal, and the working relationship could not have been better. Minna Resnick designed a stunning cover and logo. Cornell's Kathy King probably does not realize how much cumu-lative help she intermittently provided in turning scribbled lecture notes into something I could feed the TEXmachine. Richard Davis (Colorado State University), Gennady Sammorodnitsky (CorneU) and Richard Ser-fozo (Georgia Institute of Technology) used the manuscript in classroom settings and provided extensive lists of corrections and perceptive sugges-tious. A mature student perspective was provided by David Lando (Cor-nell) who read almost the whole manuscript and made an uncountable number of amazingly wise suggestions about organization and presenta-tion, as well as finding his quota of mistakes. Douglas McBeth made useful comments about appropriate levels of presentation and numerical issues.David Lando and Eleftherios Iakavou helped convince me that Harry could become friends with students whose mother tongue was different from Eng-lish. Joan Lieberman convinced me even a lawyer could appreciate Harry.Minna, Rachel and Nathan Resnick provided a warm, loving family life and generously shared the home computer with me. They were also very consoling as I coped with two hard disk crashes and a monitor melt-down.
　　 While writing a previous book in 1985, I wore out two mechanical pen-cils. The writing of this book took place on four different computers.Financial support for modernizing the computer equipment came from the National Science Foundation, Cornell's Mathematical Sciences Institute and Cornell's School of Operations Research and Industrial Engineering.Having new equipment postponed the arrival of bifocals and made that marvellous tool called TEX almost fun to use.