Probability theory : a comprehensive course / Achim Klenke.

Format:

Book

Author/Creator:

Klenke, Achim.

Series:

Universitext

Standardized Title:

Wahrscheinlichkeitstheorie. English

Language:

English

German

Subjects (All):

Probabilities.

Physical Description:

xii, 616 pages : illustrations ; 24 cm.

Place of Publication:

New York ; London : Springer, 2008.

Language Note:

Translated from the German language edition originally published by Springer, Berlin, 2006.

Summary:

Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.

This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as: limit theorems for sums of random variables; martingales; percolation; Markov chains and electrical networks; construction of stochastic processes; Poisson point processes and infinite divisibility; large deviation principles and statistical physics; Brownian motion; and stochastic integral and stochastic differential equations.

The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.

Contents:

1 Basic Measure Theory 1

1.1 Classes of Sets 1

1.2 Set Functions 12

1.3 The Measure Extension Theorem 18

1.4 Measurable Maps 34

1.5 Random Variables 43

2 Independence 49

2.1 Independence of Events 49

2.2 Independent Random Variables 56

2.3 Kolmogorov's 0-1 Law 63

2.4 Example: Percolation 66

3 Generating Functions 77

3.1 Definition and Examples 77

3.2 Poisson Approximation 80

3.3 Branching Processes 82

4 The Integral 85

4.1 Construction and Simple Properties 85

4.2 Monotone Convergence and Fatou's Lemma 93

4.3 Lebesgue Integral versus Riemann Integral 95

5 Moments and Laws of Large Numbers 101

5.1 Moments 101

5.2 Weak Law of Large Numbers 108

5.3 Strong Law of Large Numbers 111

5.4 Speed of Convergence in the Strong LLN 119

5.5 The Poisson Process 123

6 Convergence Theorems 129

6.1 Almost Sure and Measure Convergence 129

6.2 Uniform Integrability 134

6.3 Exchanging Integral and Differentiation 140

7 L[superscript p]-Spaces and the Radon-Nikodym Theorem 143

7.2 Inequalities and the Fischer-Riesz Theorem 145

7.3 Hilbert Spaces 151

7.4 Lebesgue's Decomposition Theorem 154

7.5 Supplement: Signed Measures 158

7.6 Supplement: Dual Spaces 165

8 Conditional Expectations 169

8.1 Elementary Conditional Probabilities 169

8.2 Conditional Expectations 173

8.3 Regular Conditional Distribution 179

9 Martingales 189

9.1 Processes, Filtrations, Stopping Times 189

9.2 Martingales 194

9.3 Discrete Stochastic Integral 198

9.4 Discrete Martingale Representation Theorem and the CRR Model 200

10 Optional Sampling Theorems 205

10.1 Doob Decomposition and Square Variation 205

10.2 Optional Sampling and Optional Stopping 209

10.3 Uniform Integrability and Optional Sampling 214

11 Martingale Convergence Theorems and Their Applications 217

11.1 Doob's Inequality 217

11.2 Martingale Convergence Theorems 219

11.3 Example: Branching Process 228

12 Backwards Martingales and Exchangeability 231

12.1 Exchangeable Families of Random Variables 231

12.2 Backwards Martingales 236

12.3 De Finetti's Theorem 239

13 Convergence of Measures 245

13.1 A Topology Primer 245

13.2 Weak and Vague Convergence 251

13.3 Prohorov's Theorem 259

13.4 Application: A Fresh Look at de Finetti's Theorem 268

14 Probability Measures on Product Spaces 271

14.1 Product Spaces 272

14.2 Finite Products and Transition Kernels 275

14.3 Kolmogorov's Extension Theorem 283

14.4 Markov Semigroups 288

15 Characteristic Functions and the Central Limit Theorem 293

15.1 Separating Classes of Functions 293

15.2 Characteristic Functions: Examples 300

15.3 Levy's Continuity Theorem 307

15.4 Characteristic Functions and Moments 312

15.5 The Central Limit Theorem 317

15.6 Multidimensional Central Limit Theorem 324

16 Infinitely Divisible Distributions 327

16.1 Levy-Khinchin Formula 327

16.2 Stable Distributions 339

17 Markov Chains 345

17.1 Definitions and Construction 345

17.2 Discrete Markov Chains: Examples 352

17.3 Discrete Markov Processes in Continuous Time 356

17.4 Discrete Markov Chains: Recurrence and Transience 361

17.5 Application: Recurrence and Transience of Random Walks 365

17.6 Invariant Distributions 372

18 Convergence of Markov Chains 379

18.1 Periodicity of Markov Chains 379

18.2 Coupling and Convergence Theorem 383

18.3 Markov Chain Monte Carlo Method 390

18.4 Speed of Convergence 398

19 Markov Chains and Electrical Networks 403

19.1 Harmonic Functions 404

19.2 Reversible Markov Chains 407

19.3 Finite Electrical Networks 408

19.4 Recurrence and Transience 414

19.5 Network Reduction 421

19.6 Random Walk in a Random Environment 427

20 Ergodic Theory 431

20.2 Ergodic Theorems 435

20.4 Application: Recurrence of Random Walks 439

20.5 Mixing 442

21 Brownian Motion 447

21.1 Continuous Versions 447

21.2 Construction and Path Properties 454

21.3 Strong Markov Property 459

21.4 Supplement: Feller Processes 462

21.5 Construction via L[superscript 2]-Approximation 465

21.6 The Space C([0, [infinity])) 469

21.7 Convergence of Probability Measures on C([0, [infinity])) 471

21.8 Donsker's Theorem 474

21.9 Pathwise Convergence of Branching Processes* 477

21.10 Square Variation and Local Martingales 483

22 Law of the Iterated Logarithm 495

22.1 Iterated Logarithm for the Brownian Motion 495

22.2 Skorohod's Embedding Theorem 498

22.3 Hartman-Wintner Theorem 503

23 Large Deviations 505

23.1 Cramer's Theorem 506

23.2 Large Deviations Principle 510

23.3 Sanov's Theorem 514

23.4 Varadhan's Lemma and Free Energy 519

24 The Poisson Point Process 525

24.1 Random Measures 525

24.2 Properties of the Poisson Point Process 529

24.3 The Poisson-Dirichlet Distribution* 535

25 The Ito Integral 543

25.1 Ito Integral with Respect to Brownian Motion 543

25.2 Ito Integral with Respect to Diffusions 551

25.3 The Ito Formula 554

25.4 Dirichlet Problem and Brownian Motion 562

25.5 Recurrence and Transience of Brownian Motion 564

26 Stochastic Differential Equations 567

26.1 Strong Solutions 567

26.2 Weak Solutions and the Martingale Problem 576

26.3 Weak Uniqueness via Duality 583.

Notes:

Includes bibliographical references (pages [591]-598) and indexes.

ISBN:

1848000472

9781848000476

1848000480

9781848000483

OCLC:

176833294