Foundations of Modern Probability

Frontmatter

Chapter 1. Elements of Measure Theory

Abstract

σ-fields and monotone classes; measurable functions; measures and integration; monotone and dominated convergence; transformation of integrals; product measures and Fubini’s theorem; L^p-spaces and projection; measure spaces and kernels

Chapter 2. Processes, Distributions, and Independence

Abstract

Random elements and processes; distributions and expectation; independence; zero-one laws; Borel-Cantelli lemma; Bernoulli sequences and existence; moments and continuity of paths

Chapter 3. Random Sequences, Series, and Averages

Abstract

Convergence in probability and in L^p; uniform integrability and tightness; convergence in distribution; convergence of random series; strong laws of large numbers; Portmanteau theorem; continuous mapping and approximation; coupling and measurability

Chapter 4. Characteristic Functions and Classical Limit Theorems

Abstract

Uniqueness and continuity theorem; Poisson convergence; positive and symmetric terms; Lindeberg’s condition; general Gaussian convergence; weak laws of large numbers; domain of Gaussian attraction; vague and weak compactness

Chapter 5. Conditioning and Disintegration

Abstract

Conditional expectations and probabilities; regular conditional distributions; disintegration theorem; conditional independence; transfer and coupling; Daniell-Kolmogorov theorem; extension by conditioning

Chapter 6. Martingales and Optional Times

Abstract

Filtrations and optional times; random time-change; martingale property; optional stopping and sampling; maximum and upcrossing inequalities; martingale convergence, regularity, and closure; limits of conditional expectations; regularization of submartingales

Chapter 7. Markov Processes and Discrete-Time Chains

Abstract

Markov property and transition kernels; finite-dimensional distributions and existence; space homogeneity and independence of increments; strong Markov property and excursions; invariant distributions and stationarity; recurrence and transience; ergodic behavior of irreducible chains; mean recurrence times

Chapter 8. Random Walks and Renewal Theory

Abstract

Recurrence and transience; dependence on dimension; general recurrence criteria; symmetry and duality; Wiener-Hopf factorization; ladder time and height distribution; stationary renewal process; renewal theorem

Chapter 9. Stationary Processes and Ergodic Theory

Abstract

Stationarity, invariance, and ergodicity; mean and a.s. ergodic theorem; continuous time and higher dimensions; ergodic decomposition; subadditive ergodic theorem; products of random matrices; exchangeable sequences and processes; predictable sampling

Chapter 10. Poisson and Pure Jump-Type Markov Processes

Abstract

Existence and characterizations of Poisson processes; Cox processes, randomization and thinning; one-dimensional uniqueness criteria; Markov transition and rate kernels; embedded Markov chains and explosion; compound and pseudo-Poisson processes; Kolmogorov’s backward equation; ergodic behavior of irreducible chains

Chapter 11. Gaussian Processes and Brownian Motion

Abstract

Symmetries of Gaussian distribution; existence and path properties of Brownian motion; strong Markov and reflection properties; arcsine and uniform laws; law of the iterated logarithm; Wiener integrals and isonormal Gaussian processes; multiple Wiener-Itô integrals; chaos expansion of Brownian functionals

Chapter 12. Skorohod Embedding and Invariance Principles

Abstract

Embedding of random variables; approximation of random walks; functional central limit theorem; law of the iterated logarithm; arcsine laws; approximation of renewal processes; empirical distribution functions; embedding and approximation of martingales

Chapter 13. Independent Increments and Infinite Divisibility

Abstract

Regularity and jump structure; Lévy representation; independent increments and infinite divisibility; stable processes; characteristics and convergence criteria; approximation of Lévy processes and random walks; limit theorems for null arrays; convergence of extremes

Chapter 14. Convergence of Random Processes, Measures, and Sets

Abstract

Relative compactness and tightness; uniform topology on C(K,S); Skorohod’s J₁-topology; equicontinuity and tightness; convergence of random measures; superposition and thinning; exchangeable sequences and processes; simple point processes and random closed sets

Chapter 15. Stochastic Integrals and Quadratic Variation

Abstract

Continuous local martingales and semimartingales; quadratic variation and covariation; existence and basic properties of the integral; integration by parts and Itô’s formula; Fisk-Stratonovich integral; approximation and uniqueness; random time-change; dependence on parameter

Chapter 16. Continuous Martingales and Brownian Motion

Abstract

Martingale characterization of Brownian motion; random time-change of martingales; isotropic local martingales; integral representations of martingales; iterated and multiple integrals; change of measure and Girsanov’s theorem; Cameron-Martin theorem; Wald’s identity and Novikov’s condition

Chapter 17. Feller Processes and Semigroups

Abstract

Semigroups, resolvents, and generators; closure and core; Hille-Yosida theorem; existence and regularization; strong Markov property; characteristic operator; diffusions and elliptic operators; convergence and approximation

Chapter 18. Stochastic Differential Equations and Martingale Problems

Abstract

Linear equations and Ornstein-Uhlenbeck processes; strong existence, uniqueness, and nonexplosion criteria; weak solutions and local martingale problems; well-posedness and measurability; pathwise uniqueness and functional solution; weak existence and continuity; transformation of SDEs; strong Markov and Feller properties

Chapter 19. Local Time, Excursions, and Additive Functionals

Abstract

Tanaka’s formula and semimartingale local time; occupation density, continuity and approximation; regenerative sets and processes; excursion local time and Poisson process; Ray-Knight theorem; excessive functions and additive functionals; local time at regular point; additive functionals of Brownian motion

Chapter 20. One-Dimensional SDEs and Diffusions

Abstract

Weak existence and uniqueness; pathwise uniqueness and comparison; scale function and speed measure; time-change representation; boundary classification; entrance boundaries and Feller properties; ratio ergodic theorem; recurrence and ergodicity

Chapter 21. PDE-Connections and Potential Theory

Abstract

Backward equation and Feynman-Kac formula; uniqueness for SDEs from existence for PDEs; harmonic functions and Dirichlet’s problem; Green functions as occupation densities; sweeping and equilibrium problems; dependence on conductor and domain; time reversal; capacities and random sets

Chapter 22. Predictability, Compensation, and Excessive Functions

Abstract

Accessible and predictable times; natural and predictable processes; Doob-Meyer decomposition; quasi-left-continuity; compensation of random measures; excessive and superharmonic functions; additive functionals as compensators; Riesz decomposition

Chapter 23. Semimartingales and General Stochastic Integration

Abstract

Predictable covariation and L²-integral; semimartingale integral and covariation; general substitution rule; Doléans’ exponential and change of measure; norm and exponential inequalities; martingale integral; decomposition of semimartingales; quasi-martingales and stochastic integrators

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