Adventures in Stochastic Processes

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Springer Science & Business Media, 1992. 9. 3. - 626페이지
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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.

 

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Preliminaries Discrete Index Sets andor Discrete State Spaces
1
12 CONVOLUTION
5
13 GENERATING FUNCTIONS
7
131 DIFFERENTIATION OF GENERATING FUNCTIONS
9
132 GENERATING FUNCTIONS AND MOMENTS
10
133 GENERATING FUNCTIONS AND CONVOLUTION
12
134 GENERATING FUNCTIONS COMPOUNDING AND RANDOM SUMS
15
14 THE SIMPLE BRANCHING PROCESS
18
43 TRANSFORMING POISSON PROCESSES
308
431 MAXSTABLE AND STABLE RANDOM VARIABLES
313
44 MORE TRANSFORMATION THEORY MARKING AND THINNING
316
45 THE ORDER STATISTIC PROPERTY
321
46 VARIANTS OF THE POISSON PROCESS
327
47 TECHNICAL BASICS
333
471 THE LAPLACE FUNCTIONAL
336
48 MORE ON THE POISSON PROCESS
337

15 LIMIT DISTRIBUTIONS AND THE CONTINUITY THEOREM
27
151 THE LAW OF RARE EVENTS
30
16 THE SIMPLE RANDOM WALK
33
17 THE DISTRIBUTION OF A PROCESS
40
18 STOPPING TIMES
44
181 WALDS IDENTITY
47
182 SPLITTING AN IID SEQUENCE AT A STOPPING TIME
48
EXERCISES
51
Markov Chains
60
21 CONSTRUCTION AND FIRST PROPERTIES
61
22 EXAMPLES
66
23 HIGHER ORDER TRANSITION PROBABILITIES
72
24 DECOMPOSITION OF THE STATE SPACE
77
25 THE DISSECTION PRINCIPLE
81
26 TRANSIENCE AND RECURRENCE
85
27 PERIODICITY
91
28 SOLIDARITY PROPERTIES
92
29 EXAMPLES
94
210 CANONICAL DECOMPOSITION
98
211 ABSORPTION PROBABILITIES
102
212 INVARIANT MEASURES AND STATIONARY DISTRIBUTIONS
116
2121 TIME AVERAGES
122
213 LIMIT DISTRIBUTIONS
126
2131 MORE ON NULL RECURRENCE AND TRANSIENCE
134
214 COMPUTATION OF THE STATIONARY DISTRIBUTION
137
215 CLASSIFICATION TECHNIQUES
142
EXERCISES
147
Renewal Theory
174
32 ANALYTIC INTERLUDE
176
322 CONVOLUTION
178
323 LAPLACE TRANSFORMS
181
33 COUNTING RENEWALS
185
34 RENEWAL REWARD PROCESSES
192
35 THE RENEWAL EQUATION
197
351 RISK PROCESSES
205
36 THE POISSON PROCESS AS A RENEWAL PROCESS
211
37 AN INFORMAL DISCUSSION OF RENEWAL LIMIT THEOREMS AND REGENERATIVE PROCESSES
212
371 AN INFORMAL DISCUSSION OF REGENERATIVE PROCESSES
215
38 DISCRETE RENEWAL THEORY
221
39 STATIONARY RENEWAL PROCESSES
224
310 THE BLACKWELL AND KEY RENEWAL THEOREMS
230
3101 DIRECT RIEMANN INTEGRABILITY
231
3102 EQUIVALENT FORMS OF THE RENEWAL THEOREM
237
3103 PROOF OF THE RENEWAL THEOREM
243
311 IMPROPER RENEWAL EQUATIONS
253
312 MORE ON REGENERATIVE PROCESSES
259
3122 THE RENEWAL EQUATION AND SMITHS THEOREM
263
3123 QUEUEING EXAMPLES
269
EXERCISES
280
Point Processes
300
42 THE POISSON PROCESS
303
49 A GENERAL CONSTRUCTION OF THE POISSON PROCESS A SIMPLE DERIVATION OF THE ORDER STATISTIC PROPERTY
341
410 MORE TRANSFORMATION THEORY LOCATION DEPENDENT THINNING
343
411 RECORDS
346
EXERCISES
349
Continuous Time Markov Chains
367
52 STABILITY AND EXPLOSIONS
375
521 THE MARKOV PROPERTY
377
53 DISSECTION
380
54 THE BACKWARD EQUATION AND THE GENERATOR MATRIX
382
55 STATIONARY AND LIMITING DISTRIBUTIONS
392
551 MORE ON INVARIANT MEASURES
398
56 LAPLACE TRANSFORM METHODS
402
57 CALCULATIONS AND EXAMPLES
406
571 QUEUEING NETWORKS
415
58 TIME DEPENDENT SOLUTIONS
426
59 REVERSIBILITY
431
510 UNIFORMIZABLE CHAINS
436
511 THE LINEAR BIRTH PROCESS AS A POINT PROCESS
439
EXERCISES
446
Brownian Motion
482
62 PRELIMINARIES
487
63 CONSTRUCTION OF BROWNIAN MOTION
489
64 SIMPLE PROPERTIES OF STANDARD BROWNIAN MOTION
494
65 THE REFLECTION PRINCIPLE AND THE DISTRIBUTION OF THE MAXIMUM
497
66 THE STRONG INDEPENDENT INCREMENT PROPERTY AND REFLECTION
504
67 ESCAPE FROM A STRIP
508
68 BROWNIAN MOTION WITH DRIFT
511
69 HEAVY TRAFFIC APPROXIMATIONS IN QUEUEING THEORY
514
610 THE BROWNIAN BRIDGE AND THE KOLMOGOROVSMIRNOV STATISTIC
524
611 PATH PROPERTIES
539
612 QUADRATIC VARIATION
542
613 KHINTCHINES LAW OF THE ITERATED LOGARITHM FOR BROWNIAN MOTION
546
EXERCISES
551
The General Random Walk
558
71 STOPPING TIMES
559
72 GLOBAL PROPERTIES
561
PROBABILISTIC INTERPRETATIONS OF TRANSFORMS
564
74 DUAL PAIRS OF STOPPING TIMES
568
75 WIENERHOPF DECOMPOSITIONS
573
76 CONSEQUENCES OF THE WIENERHOPF FACTORIZATION
581
77 THE MAXIMUM OF A RANDOM WALK
587
78 RANDOM WALKS AND THE GG1 QUEUE
591
781 EXPONENTIAL RIGHT TAIL
595
782 APPLICATIONS TO THE GM1 QUEUEING MODEL
599
783 EXPONENTIAL LEFT TAIL
602
784 THE MG1 QUEUE
605
785 QUEUE LENGTHS
607
Bibliography
613
Index
617
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