A Modern Approach to Probability Theory

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Springer Science & Business Media, Dec 23, 1996 - Mathematics - 758 pages
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Overview This book is intended as a textbook in probability for graduate students in math ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri butions, and minimum variance unbiased estimation.
 

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Contents

Chapter 1 Probability Spaces
3
12 Ingredients of probability spaces
6
13 𝝈fields
8
14 Borel 𝝈fields
9
Chapter 2 Random Variables
11
22 Rdvalued random variables
14
23 Rvalued random variables
18
24 Further examples
20
191 Certain sequences of distributions on C0 1
368
192 The existence of and convergence to Wiener measure
371
193 Some measurable functionals on C01
374
194 Drownian motion on 0 oc
379
195 Filtrations and stopping times
382
196 Brownian motion nitrations and stopping times
385
197 Characteri2ation of Brownian motion
389
198 Law of the Iterated Logarithm
390

Chapter 3 Distribution Functions
25
32 Examples of distributions
29
33 Some descriptive terminology
31
34 Distributions with densities
35
35 Further examples
37
36 Distribution functions for the extended real line
39
Theory
41
42 Linearity and positivity
46
43 Monotone convergence
49
44 Expectation of compositions
51
45 The RiemannStieltjes integral and expectations
53
Applications
59
52 Mean vectors and covariance matrices
66
53 Moments and the Jensen Inequality
68
54 Probability generating functions
70
55 Characterization of probability generating functions
73
Chapter 6 Calculating Probabilities and Measures
75
62 The BorelCantelli and KochenStone Lemmas
77
63 Inclusionexclusion
80
64 Finite and afinite measures
81
Existence and Uniqueness
85
72 Finitely additive functions defined on fields
87
73 Existence extension and completion of measures
90
74 Examples
95
Chapter 8 Integration Theory
101
82 Convergence theorems
105
83 Probability measures and infinite measures compared
110
84 Lebesgue integrals and RiemannStieltjes integrals
111
85 Absolute continuity and densities
115
86 Integration with respect to counting measure
118
Chapter 9 Stochastic Independence
121
finitely many factors
127
93 The Fubini Theorem
130
94 Expectations and independence
133
95 Densities and independence
134
infinitely many factors
136
97 The BorelCantelli Lemma and independent sequences
141
98 Order statistics
143
99 Some new distributions involving independence
145
Chapter 10 Sums of Independent Random Variables
147
102 Multinomial distributions
152
103 Probability generating functions and sums in Z
153
104 Dirichlet distributions
156
105 Random sums in various settings
159
Chapter 11 Random Walk
163
112 Definition and examples
164
113 Filtrations and stopping times
171
114 Stopping limes and random walks
174
115 A hittingtime example
175
116 Returns to 0
178
117 Random walks in various settings
183
Chapter 12 Theorems of AS Convergence
185
122 Laws of Large Numbers
188
123 Applications
193
124 01 laws In the Strong Law of Large Numbers we are interested in the event
196
125 Random infinite series
198
126 The Etemadi Lemma
200
127 The Kolmogorov ThreeSeries Theorem
202
128 The image of a random walk
206
Chapter 13 Characteristic Functions
209
132 The Parseval Relation and uniqueness
211
133 Characteristic functions of convolutions
214
134 Symmetrization
217
135 Moment generating functions
218
136 Moment theorems
223
137 Inversion theorems
226
138 Characteristic functions in Rd
234
139 Normal distributions on ddimensional space
237
1310 An application to random walks on Z
238
1311 An application to the calculation of a sum
239
Chapter 14 Convergence in Distribution on the Real Line
243
142 Limit distributions for extreme values
246
143 Relationships to other types of convergence
249
144 Convergence conditions for sequences of distributions
252
145 Sequences of distributions on R
254
146 Relative sequential compactness
255
147 The Continuity Theorem
260
148 Scaling and centering of sequences of distributions
263
149 Characteri2ation of moment generating functions
266
1410 Characterization of characteristic functions
269
Chapter 15 Distributional Limit Theorems for Partial Sums
271
151 Infinite series of independent random variables
272
152 The Law of Large Numbers revisited
273
153 The Classical Central Limit Theorem
275
154 The general setting for iid sequences
277
155 Large deviations
280
156 Local limit theorems
282
Chapter 16 Infinitely Divisible Distributions as Limits
289
162 Infinitely divisible distributions on R
293
163 LÚvyKhinchin representations
295
164 Infinitely divisible distributions on R+
300
165 Extension to R
302
introduction
303
167 lid triangular arrays
306
168 Symmetric and nonnegative triangular arrays
310
169 General triangular arrays
313
Chapter 17 Stable Distributions as Limits
323
171 Regular variation
324
172 The stable distributions
326
173 Domains of attraction
332
174 Domains of strict attraction
342
Chapter 18 Convergence in Distribution on Polish Spaces
347
182 Definition of and criteria for convergence
352
183 Relative sequential compactness
355
184 Uniform tightness and the Prohorov Theorem
357
185 Convergence in product spaces
358
186 The Continuity Theorem for Rd
361
187 The Prohorov metric
363
Chapter 19 The Invariance Principle and Brownian Motion
367
Chapter 20 Spaces of Random Variables
395
202 The Hilbert space L2ΩFP
397
203 The metric space L1ΩFP
401
204 Best linear estimator
402
Chapter 21 Conditional Probabilities
403
212 Conditional distributions
412
213 Conditional densities
416
214 Existence and uniqueness of conditional distributions
417
215 Conditional independence
422
216 Conditional distributions of normal random vectors
427
Chapter 22 Construction of Random Sequences
429
222 Construction of exchangeable sequences
433
223 Construction of Markov sequences
436
224 Polya urns
437
225 Coupon collecting
439
Chapter 23 Conditional Expectations
443
232 Conditional versions of unconditional theorems
448
233 Formulas for conditional expectations
451
234 Conditional variance
453
Chapter 24 Martingales
459
242 Examples
461
243 Doob decomposition
465
244 Transformations of submartingales
466
optional sampling
467
246 Applications of optional sampling
473
247 Inequalities and convergence results
477
248 Optimal strategy in Red and Black
484
Chapter 25 Renewal Sequences
489
251 Basic criterion
490
252 Renewal measures and potential measures
491
253 Examples
494
a first step
497
255 Delayed renewal sequences
499
256 The Renewal Theorem
502
257 Applications to random walks
506
Chapter 26 Timehomogeneous Markov Sequences
511
262 Examples
515
263 Martingales and the strong Markov property
520
264 Hitting times and return times
522
265 Renewal theory and Markov sequences
525
266 Irreducible Markov sequences
527
267 Equilibrium distributions
528
Chapter 27 Exchangeable Sequences
533
272 Infinite exchangeable sequences
539
273 Posterior distributions
542
274 Generali2ation to Borel spaces
544
275 Ferguson distributions and BlackwellMacQueen urns
550
Chapter 28 Stationary Sequences
553
282 Notation
555
283 Examples
556
284 The Birkhoff Ergodic Theorem
558
285 Ergodicity
561
286 The KingmanLiggett Subadditive Ergodic Theorem
564
287 Spectral analysis of stationary sequences
571
Chapter 29 Point Processes
581
292 Intensity measures
586
293 Poisson point processes
587
294 Examples of Poisson point processes
589
295 Probability generating functionals
594
296 Operations on point processes
597
297 Convergence in distribution for point processes
598
Chapter 30 Levy Processes
601
302 Definition of Levy process
602
303 Construction of Levy processes
605
304 Filtrations and stopping times
610
305 Subordination
611
306 Localtime processes and regenerative subsets of 0 oo
612
307 Sample function properties of subordinators
618
Chapter 31 Introduction to Markov Processes
621
312 Markov strong Markov and Feller processes
622
313 Infinitesimal generators
628
314 The martingale problem We adapt the main definition of Chapter 24 to the continuoustime setting
629
bounded rates
632
unbounded rates
636
317 Renewal theory for purejump Markov processes
639
Chapter 32 Interacting Particle Systems
641
322 The universal coupling
644
323 Examples
651
324 Equilibrium distributions
655
325 Systems with attractive infinitesimal generators
657
Chapter 33 DiIfusions and Stochastic Calculus
661
332 The It˘ integral
663
333 Stochastic diIferentials and the Ito Lemma
668
334 Autonomous stochastic differential equations
672
335 Generators and the Dirichlet problem
679
336 Diffusions in higher dimensions
682
APPENDIX A Notation and Usage of Terms
687
A 2 Usage Binary digits is used for what some call bits
690
A3 Exercises on subtle distinctions
692
APPENDIX B Metric Spaces
693
B2 Sequences
694
B3 Continuous functions
695
APPENDIX C Topological Spaces
697
C2 Compactification
699
C3 Product topologies
700
C5 Limits and continuous functions
701
APPENDIX D RiemannStieltjes Integration
703
D2 Relation to the Riemann integral
705
D3 Change of variables
706
D4 Integration by parts
707
D5 Improper RiemannStieltjes integrals
708
Appendix E Taylor Approximations CValued Logarithms
709
E2 Complex exponentials and logarithms
711
E3 Approximations of general Cvalued functions
714
APPENDIX F Bibliography
715
APPENDIX G Comments and Credits
723
Index
737
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A Probability Path
Sidney Resnick
Limited preview - 2003
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