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Probability with Martingales

Book Description

Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface – please read!
  6. A Question of Terminology
  7. A Guide to Notation
  8. Chapter 0: A Branching-Process Example
    1. 0.0. Introductory remarks
    2. 0.1. Typical number of children, X
    3. 0.2. Size of nth generation, Zn
    4. 0.3. Use of conditional expectations
    5. 0.4. Extinction probability, π
    6. 0.5. Pause for thought: measure
    7. 0.6. Our first martingale
    8. 0.7. Convergence (or not) of expectations
    9. 0.8. Finding the distribution of M∞
    10. 0.9. Concrete example
  9. Part A: Foundations
    1. Chapter 1: Measure Spaces
      1. 1.0. Introductory remarks
      2. 1.1. Definitions of algebra, σ-algebra
      3. 1.2. Examples. Borel σ-algebras, β(S), β = β(R)
      4. 1.3. Definitions concerning set functions
      5. 1.4. Definition of measure space
      6. 1.5. Definitions concerning measures
      7. 1.6. Lemma. Uniqueness of extension, π-systems
      8. 1.7. Theorem. Carathéodory’s extension theorem
      9. 1.8. Lebesgue measure Leb on ((0, l], औ(0,1])
      10. 1.9. Lemma. Elementary inequalities
      11. 1.10. Lemma. Monotone-convergence properties of measures
      12. 1.11. Example/Warning
    2. Chapter 2: Events
      1. 2.1. Model for experiment: (Ω, , P)
      2. 2.2. The intuitive meaning
      3. 2.3. Examples of (Ω, ) pairs
      4. 2.4. Almost surely (a.s.)
      5. 2.5. Reminder: lim sup, lim inf, ↓ lim, etc
      6. 2.6. Definitions. limsup En, (En, i.o.)
      7. 2.7. First Borel-Cantelli Lemma (BC1)
      8. 2.8. Definitions, lim inf En, (En, ev)
      9. 2.9. Exercise
    3. Chapter 3: Random Variables
      1. 3.1. Definitions. Σ-measurable function, mΣ, (mΣ)+, bΣ
      2. 3.2. Elementary Propositions on measurability
      3. 3.3. Lemma. Sums and products of measurable functions are measurable
      4. 3.4. Composition Lemma
      5. 3.5. Lemma on measurability of infs, lim infs of functions
      6. 3.6. Definition. Random variable
      7. 3.7. Example. Coin tossing
      8. 3.8. Definition, σ-algebra generated by a collection of functions on Ω
      9. 3.9. Definitions. Law, Distribution Function
      10. 3.10. Properties of distribution functions
      11. 3.11. Existence of random variable with given distribution function
      12. 3.12. Skorokod representation of a random variable with prescribed distribution function
      13. 3.13. Generated σ-algebras – a discussion
      14. 3.14. The Monotone-Class Theorem
    4. Chapter 4: Independence
      1. 4.1. Definitions of independence
      2. 4.2. The π-system Lemma; and the more familiar definitions
      3. 4.3. Second Borel-Cantelli Lemma (BC2)
      4. 4.4. Example
      5. 4.5. A fundamental question for modelling
      6. 4.6. A coin-tossing model with applications
      7. 4.7. Notation: IID RVs
      8. 4.8. Stochastic processes; Markov chains
      9. 4.9. Monkey typing Shakespeare
      10. 4.10. Definition. Tail σ-algebras
      11. 4.11. Theorem. Kolmogorov’s 0-1 law
      12. 4.12. Exercise/Warning
    5. Chapter 5: Integration
      1. 5.0. Notation, etc. μ(f) :=: ∫ fdμ, μ(f; A)
      2. 5.1. Integrals of non-negative simple functions, SF+
      3. 5.2. Definition of μ(f), f ∈ (mΣ)+
      4. 5.3. Monotone-Convergence Theorem (MON)
      5. 5.4. The Fatou Lemmas for functions (FA-TOU)
      6. 5.5. ‘Linearity’
      7. 5.6. Positive and negative parts of f
      8. 5.7. Integrable function, (S, Σ, μ)
      9. 5.8. Linearity
      10. 5.9. Dominated Convergence Theorem (DOM)
      11. 5.10. Scheffe’s Lemma (SCHEFFÉ)
      12. 5.11. Remark on uniform integrability
      13. 5.12. The standard machine
      14. 5.13. Integrals over subsets
      15. 5.14. The measure fμ, f ∈ (mΣ)+
    6. Chapter 6: Expectation
      1. 6.1. Definition of expectation
      2. 6.2. Convergence theorems
      3. 6.3. The notation E(X; F)
      4. 6.4. Markov’s inequality
      5. 6.5. Sums of non-negative RVs
      6. 6.6. Jensen’s inequality for convex functions
      7. 6.7. Monotonicity of norms
      8. 6.8. The Schwarz inequality
      9. 6.9. L2: Pythagoras, covariance, etc
      10. 6.10. Completeness of (1 ≤ p ≤ ∞)
      11. 6.11. Orthogonal projection
      12. 6.12. The ‘elementary formula’ for expectation
      13. 6.13. Hölder from Jensen
    7. Chapter 7: An Easy Strong Law
      1. 7.1. ‘Independence means multiply’ – again!
      2. 7.2. Strong Law – first version
      3. 7.3. Chebyshev’s inequality
      4. 7.4. Weierstrass approximation theorem
    8. Chapter 8: Product Measure
      1. 8.0. Introduction and advice
      2. 8.1. Product measurable structure, Σ1 × Σ2
      3. 8.2. Product measure, Fubini’s Theorem
      4. 8.3. Joint laws, joint pdfs
      5. 8.4. Independence and product measure
      6. 8.5. औ(R)n = औ(Rn)
      7. 8.6. The n-fold extension
      8. 8.7. Infinite products of probability triples
      9. 8.8. Technical note on the existence of joint laws
  10. Part B: Martingale Theory
    1. Chapter 9: Conditional Expectation
      1. 9.1. A motivating example
      2. 9.2. Fundamental Theorem and Definition (Kolmogorov, 1933)
      3. 9.3. The intuitive meaning
      4. 9.4. Conditional expectation as least-squares-best predictor
      5. 9.5. Proof of Theorem 9.2
      6. 9.6. Agreement with traditional expression
      7. 9.7. Properties of conditional expectation: a list
      8. 9.8. Proofs of the properties in Section 9.7
      9. 9.9. Regular conditional probabilities and pdfs
      10. 9.10. Conditioning under independence assumptions
      11. 9.11. Use of symmetry: an example
    2. Chapter 10: Martingales
      1. 10.1. Filtered spaces
      2. 10.2. Adapted processes
      3. 10.3. Martingale, super-martingale, submartingale
      4. 10.4. Some examples of martingales
      5. 10.5. Fair and unfair games
      6. 10.6. Previsible process, gambling strategy
      7. 10.7. A fundamental principle: you can’t beat the system!
      8. 10.8. Stopping time
      9. 10.9. Stopped supermartingales are supermartingales
      10. 10.10. Doob’s Optional-Stopping Theorem
      11. 10.11. Awaiting the almost inevitable
      12. 10.12. Hitting times for simple random walk
      13. 10.13. Non-negative superharmonic functions for Markov chains
    3. Chapter 11: The Convergence Theorem
      1. 11.1. The picture that says it all
      2. 11.2. Upcrossings
      3. 11.3. Doob’s Upcrossing Lemma
      4. 11.4. Corollary
      5. 11.5. Doob’s’Forward’Convergence Theorem
      6. 11.6. Warning
      7. 11.7. Corollary
    4. Chapter 12: Martingales bounded in L2
      1. 12.0. Introduction
      2. 12.1. Martingales in L2: orthogonality of increments
      3. 12.2. Sums of zero-mean independent random variables in L2
      4. 12.3. Random signs
      5. 12.4. A symmetrization technique: expanding the sample space
      6. 12.5. Kolmogorov’s Three-Series Theorem
      7. 12.6. Cesáro’s Lemma
      8. 12.7. Kronecker’s Lemma
      9. 12.8. A Strong Law under variance constraints
      10. 12.9. Kolmogorov’s Truncation Lemma
      11. 12.10. Kolmogorov’s Strong Law of Large Numbers (SLLN)
      12. 12.11. Doob decomposition
      13. 12.12. The angle-brackets process M
      14. 12.13. Relating convergence of M to finiteness of M∞
      15. 12.14. A trivial ‘Strong Law’ for martingales in L2
      16. 12.15. Lévy’s extension of the Borel-Cantelli Lemmas
      17. 12.16. Comments
    5. Chapter 13: Uniform Integrability
      1. 13.1. An ‘absolute continuity’ property
      2. 13.2. Definition. UI family
      3. 13.3. Two simple sufficient conditions for the UI property
      4. 13.4. UI property of conditional expectations
      5. 13.5. Convergence in probability
      6. 13.6. Elementary proof of (BDD)
      7. 13.7. A necessary and sufficient condition for convergence
    6. Chapter 14: UI Martingales
      1. 14.0. Introduction
      2. 14.1. UI martingales
      3. 14.2. Lévy’s ‘Upward’ Theorem
      4. 14.3. Martingale proof of Kolmogorov’s 0-1 law
      5. 14.4. Lévy’s ‘Downward’ Theorem
      6. 14.5. Martingale proof of the Strong Law
      7. 14.6. Doob’s Sub-martingale Inequality
      8. 14.7. Law of the Iterated Logarithm: special case
      9. 14.8. A standard estimate on the normal distribution
      10. 14.9. Remarks on exponential bounds; large deviation theory
      11. 14.10. A consequence of Hölder’s inequality
      12. 14.11. Doob’s inequality
      13. 14.12. Kakutani’s Theorem on ‘product’ martingales
      14. 14.13.The Radon-Nikodým theorem
      15. 14.14. The Radon-Nikodým theorem and conditional expectation
      16. 14.15. Likelihood ratio; equivalent measures
      17. 14.16. Likelihood ratio and conditional expectation
      18. 14.17. Kakutani’s Theorem revisited; consistency of LR test
      19. 14.18. Note on Hardy spaces, etc
    7. Chapter 15: Applications
      1. 15.0. Introduction – please read!
      2. 15.1. A trivial martingale-representation result
      3. 15.2. Option pricing; discrete Black-Scholes formula
      4. 15.3. The Mabinogion sheep problem
      5. 15.4. Proof of Lemma 15.3(c)
      6. 15.5. Proof of result 15.3(d)
      7. 15.6. Recursive nature of conditional probabilities
      8. 15.7. Bayes’ formula for bivariate normal distributions
      9. 15.8. Noisy observation of a single random variable
      10. 15.9. The Kalman-Bucy filter
      11. 15.10. Harnesses entangled
      12. 15.11. Harnesses unravelled, 1
      13. 15.12. Harnesses unravelled, 2
  11. Part C: Characteristic Functions
    1. Chapter 16: Basic Properties of CFs
      1. 16.1. Definition
      2. 16.2. Elementary properties
      3. 16.3. Some uses of characteristic functions
      4. 16.4. Three key results
      5. 16.5. Atoms
      6. 16.6. Lévy’s Inversion Formula
      7. 16.7. A table
    2. Chapter 17: Weak Convergence
      1. 17.1. The ‘elegant’ definition
      2. 17.2. A ‘practical’ formulation
      3. 17.3. Skorokhod representation
      4. 17.4. Sequential compactness for Prob()
      5. 17.5. Tightness
    3. Chapter 18: The Central Limit Theorem
      1. 18.1. Lévy’s Convergence Theorem
      2. 18.2. o and O notation
      3. 18.3. Some important estimates
      4. 18.4. The Central Limit Theorem
      5. 18.5. Example
      6. 18.6. CF proof of Lemma 12.4
  12. Appendices
    1. Chapter A1: Appendix to Chapter 1
      1. A1.1. A non-measurable subset A of S1
      2. A1.2. d-systems
      3. A1.3. Dynkin’s Lemma
      4. A1.4. Proof of Uniqueness Lemma 1.6
      5. A1.5. λ-sets: ‘algebra’ case
      6. A1.6. Outer measures
      7. A1.7. Carathéodory’s Lemma
      8. A1.8. Proof of Carathéodory’s Theorem
      9. A1.9. Proof of the existence of Lebesgue measure on ((0,1], औ(0,1])
      10. A1.10. Example of non-uniqueness of extension
      11. A1.11. Completion of a measure space
      12. A1.12. The Baire category theorem
    2. Chapter A3: Appendix to Chapter 3
      1. A3.1. Proof of the Monotone-Class Theorem 3.14
      2. A3.2. Discussion of generated σ-algebras
    3. Chapter A4: Appendix to Chapter 4
      1. A4.1. Kolmogorov’s Law of the Iterated Logarithm
      2. A4.2. Strassen’s Law of the Iterated Logarithm
      3. A4.3. A model for a Markov chain
    4. Chapter A5: Appendix to Chapter 5
      1. A5.1. Doubly monotone arrays
      2. A5.2. The key use of Lemma 1.10(a)
      3. A5.3. ‘Uniqueness of integral’
      4. A5.4. Proof of the Monotone-Convergence Theorem
    5. Chapter A9: Appendix to Chapter 9
      1. A9.1. Infinite products: setting things up
      2. A9.2. Proof of A9.1(e)
    6. Chapter A13: Appendix to Chapter 13
      1. A13.1. Modes of convergence: definitions
      2. A13.2. Modes of convergence: relationships
    7. Chapter A14: Appendix to Chapter 14
      1. A14.1. The (σ-algebra , T a stopping time
      2. A14.2. A special case of OST
      3. A14.3. Doob’s Optional-Sampling Theorem for UI martingales
      4. A14.4. The result for UI submartingales
    8. Chapter A16: Appendix to Chapter 16
      1. A16.1. Differentiation under the integral sign
  13. Chapter E: Exercises
  14. References
  15. Index