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1. Probability and measure [2012]
 Billingsley, Patrick.
 Anniversary ed.  Hoboken, N.J. : Wiley, c2012.
 Description
 Book — xvii, 624 p. : ill ; 26 cm.
 Summary

 FOREWORD xi PREFACE xiii Patrick Billingsley
 1925
 2011 xv Chapter1 PROBABILITY
 1
 1. BOREL S NORMAL NUMBER THEOREM,
 1 The Unit Interval The Weak Law of Large Numbers The Strong Law of Large Numbers Strong Law Versus Weak Length The Measure Theory of Diophantine Approximation
 2. PROBABILITY MEASURES,
 18 Spaces Assigning Probabilities Classes of Sets Probability Measures Lebesgue Measure on the Unit Interval Sequence Space Constructing sFields
 3. EXISTENCE AND EXTENSION,
 39 Construction of the Extension Uniqueness and the p ? Theorem Monotone Classes Lebesgue Measure on the Unit Interval Completeness Nonmeasurable Sets Two Impossibility Theorems
 4. DENUMERABLE PROBABILITIES,
 53 General Formulas Limit Sets Independent Events Subfields The BorelCantelli Lemmas The ZeroOne Law
 5. SIMPLE RANDOM VARIABLES,
 72 Definition Convergence of Random Variables Independence Existence of Independent Sequences Expected Value Inequalities
 6. THE LAW OF LARGE NUMBERS,
 90 The Strong Law The Weak Law Bernstein's Theorem A Refinement of the Second BorelCantelli Lemma
 7. GAMBLING SYSTEMS,
 98 Gambler's Ruin Selection Systems Gambling Policies Bold Play Timid Play
 8. MARKOV CHAINS,
 117 Definitions HigherOrder Transitions An Existence Theorem Transience and Persistence Another Criterion for Persistence Stationary Distributions Exponential Convergence Optimal Stopping
 9. LARGE DEVIATIONS AND THE LAW OF THE ITERATED LOGARITHM,
 154 Moment Generating Functions Large Deviations Chernoff's Theorem The Law of the Iterated Logarithm Chapter2 MEASURE
 167
 10. GENERAL MEASURES,
 167 Classes of Sets Conventions Involving
 8 Measures Uniqueness
 11. OUTER MEASURE,
 174 Outer Measure Extension An Approximation Theorem
 12. MEASURES IN EUCLIDEAN SPACE,
 181 Lebesgue Measure Regularity Specifying Measures on the Line Specifying Measures in Rk Strange Euclidean Sets
 13. MEASURABLE FUNCTIONS AND MAPPINGS,
 192 Measurable Mappings Mappings into Rk Limits and Measurability Transformations of Measures
 14. DISTRIBUTION FUNCTIONS,
 198 Distribution Functions Exponential Distributions Weak Convergence Convergence of Types Extremal Distributions Chapter3 INTEGRATION
 211
 15. THE INTEGRAL,
 211 Definition Nonnegative Functions Uniqueness
 16. PROPERTIES OF THE INTEGRAL,
 218 Equalities and Inequalities Integration to the Limit Integration over Sets Densities Change of Variable Uniform Integrability Complex Functions
 17. THE INTEGRAL WITH RESPECT TO LEBESGUE MEASURE,
 234 The Lebesgue Integral on the Line The Riemann Integral The Fundamental Theorem of Calculus Change of Variable The Lebesgue Integral in Rk Stieltjes Integrals
 18. PRODUCT MEASURE AND FUBINI S THEOREM,
 245 Product Spaces Product Measure Fubini's Theorem Integration by Parts Products of Higher Order
 19. THE Lp SPACES,
 256 Definitions Completeness and Separability Conjugate Spaces Weak Compactness Some Decision Theory The Space L2 An Estimation Problem Chapter4 RANDOM VARIABLES AND EXPECTED VALUES
 271
 20. RANDOM VARIABLES AND DISTRIBUTIONS,
 271 Random Variables and Vectors Subfields Distributions Multidimensional Distributions Independence Sequences of Random Variables Convolution Convergence in Probability The GlivenkoCantelli Theorem
 21. EXPECTED VALUES,
 291 Expected Value as Integral Expected Values and Limits Expected Values and Distributions Moments Inequalities Joint Integrals Independence and Expected Value Moment Generating Functions
 22. SUMS OF INDEPENDENT RANDOM VARIABLES,
 300 The Strong Law of Large Numbers The Weak Law and Moment Generating Functions Kolmogorov's ZeroOne Law Maximal Inequalities Convergence of Random Series Random Taylor Series
 23. THE POISSON PROCESS,
 316 Characterization of the Exponential Distribution The Poisson Process The Poisson Approximation Other Characterizations of the Poisson Process Stochastic Processes
 24. THE ERGODIC THEOREM,
 330 MeasurePreserving Transformations Ergodicity Ergodicity of Rotations Proof of the Ergodic Theorem The ContinuedFraction Transformation Diophantine Approximation Chapter5 CONVERGENCE OF DISTRIBUTIONS
 349
 25. WEAK CONVERGENCE,
 349 Definitions Uniform Distribution Modulo
 1 Convergence in Distribution Convergence in Probability Fundamental Theorems Helly's Theorem Integration to the Limit
 26. CHARACTERISTIC FUNCTIONS,
 365 Definition Moments and Derivatives Independence Inversion and the Uniqueness Theorem The Continuity Theorem Fourier Series
 27. THE CENTRAL LIMIT THEOREM,
 380 Identically Distributed Summands The Lindeberg and Lyapounov Theorems Dependent Variables
 28. INFINITELY DIVISIBLE DISTRIBUTIONS,
 394 Vague Convergence The Possible Limits Characterizing the Limit
 29. LIMIT THEOREMS IN Rk,
 402 The Basic Theorems Characteristic Functions Normal Distributions in Rk The Central Limit Theorem
 30. THE METHOD OF MOMENTS,
 412 The Moment Problem Moment Generating Functions Central Limit Theorem by Moments Application to Sampling Theory Application to Number Theory Chapter6 DERIVATIVES AND CONDITIONAL PROBABILITY
 425
 31. DERIVATIVES ON THE LINE,
 425 The Fundamental Theorem of Calculus Derivatives of Integrals Singular Functions Integrals of Derivatives Functions of Bounded Variation
 32. THE RADON NIKODYM THEOREM,
 446 Additive Set Functions The Hahn Decomposition Absolute Continuity and Singularity The Main Theorem
 33. CONDITIONAL PROBABILITY,
 454 The Discrete Case The General Case Properties of Conditional Probability Difficulties and Curiosities Conditional Probability Distributions
 34. CONDITIONAL EXPECTATION,
 472 Definition Properties of Conditional Expectation Conditional Distributions and Expectations Sufficient Subfields MinimumVariance Estimation
 35. MARTINGALES,
 487 Definition Submartingales Gambling Functions of Martingales Stopping Times Inequalities Convergence Theorems Applications: Derivatives Likelihood Ratios Reversed Martingales Applications: de Finetti's Theorem Bayes Estimation A Central Limit Theorem Chapter7 STOCHASTIC PROCESSES
 513
 36. KOLMOGOROV'S EXISTENCE THEOREM,
 513 Stochastic Processes FiniteDimensional Distributions Product Spaces Kolmogorov's Existence Theorem The Inadequacy of RT A Return to Ergodic Theory The Hewitt Savage Theorem
 37. BROWNIAN MOTION,
 530 Definition Continuity of Paths Measurable Processes Irregularity of Brownian Motion Paths The Strong Markov Property The Reflection Principle Skorohod Embedding Invariance
 38. NONDENUMERABLE PROBABILITIES,
 558 Introduction Definitions Existence Theorems Consequences of Separability
 APPENDIX 571 NOTES ON THE PROBLEMS
 587 BIBLIOGRAPHY
 617 INDEX 619.
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 Online
Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

Stacks  
QA273 .B575 2012  Unknown On reserve at Li and Ma Science Library 2hour loan 
STATS310A01
 Course
 STATS310A01  Theory of Probability I
 Instructor(s)
 Montanari, Andrea
Online 2. Probability : theory and examples [2010]
 Durrett, Richard, 1951
 4th ed.  Cambridge ; New York : Cambridge University Press, 2010.
 Description
 Book — x, 428 p. : ill. ; 27 cm.
 Summary

 1. Measure theory
 2. Laws of large numbers
 3. Central limit theorems
 4. Random walks
 5. Martingales
 6. Markov chains
 7. Ergodic theorems
 8. Brownian motion Appendix A. Measure theory details.
 (source: Nielsen Book Data)
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 Also online at

 Cambridge Core Access limited to one user.
Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

Stacks  
QA273 .D865 2010  Unknown On reserve at Li and Ma Science Library 2hour loan 
STATS310A01
 Course
 STATS310A01  Theory of Probability I
 Instructor(s)
 Montanari, Andrea
3. Probability and measure [1995]
 Billingsley, Patrick.
 3rd ed.  New York : J. Wiley & Sons, c1995.
 Description
 Book — xii, 593 p. ; 25 cm.
 Summary

 Probability Measure Integration Random Variables and Expected Values Convergence of Distributions Derivatives and Conditional Probability Stochastic Processes Appendix Notes of the Problems Bibliography List of Symbols Index.
 (source: Nielsen Book Data)
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PROBABILITY AND MEASURE Third Edition Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory. Like the previous editions, this new edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.
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 Online
Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

Stacks  
QA273 .B575 1995  Unknown On reserve at Li and Ma Science Library 2hour loan 
QA273 .B575 1995  Unknown On reserve at Li and Ma Science Library 2hour loan 
STATS310A01
 Course
 STATS310A01  Theory of Probability I
 Instructor(s)
 Montanari, Andrea
4. Probability with martingales [1991]
 Williams, D. (David), 1938
 Cambridge ; New York : Cambridge University Press, c1991.
 Description
 Book — xv, 251 p. ; 23 cm.
 Summary

 1. A branchingprocess example Part I. Foundations:
 2. Measure spaces
 3. Events
 4. Random variables
 5. Independence
 6. Integration
 7. Expectation
 8. An easy strong law: product measure Part II. Martingale Theory:
 9. Conditional expectation
 10. Martingales
 11. The convergence theorem
 12. Martingales bounded in L2
 13. Uniform integrability
 14. UI martingales
 15. Applications Part III. Characteristic Functions:
 16. Basic properties of CFs
 17. Weak convergence
 18. The central limit theorem Appendices Exercises.
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 Online

 Cambridge Core Access limited to one user.
 Google Books (Full view)
Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

Stacks  
QA274.5 .W55 1991  Unknown On reserve at Li and Ma Science Library 2hour loan 
QA274.5 .W55 1991  Unknown On reserve at Li and Ma Science Library 2hour loan 
STATS310A01
 Course
 STATS310A01  Theory of Probability I
 Instructor(s)
 Montanari, Andrea