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Tables of Contents for Foundations of Modern Probability
Chapter/Section Title
Page #
Page Count
1. Elements of Measure Theory
1
21
XXX-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
2. Processes, Distributions, and Independence
22
17
random elements and processes
distributions and expectation
independence
zero-one laws
Borel-Cantelli lemma
Bernoulli sequences and existence
moments and continuity of paths
3. Random Sequences, Series, and Averages
39
21
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
4. Characteristic Functions and Classical Limit Theorems
60
20
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
5. Conditioning and Disintegration
80
16
conditional expectations and probabilities
regular conditional distributions
disintegration theorem
conditional independence
transfer and coupling
Daniell-Kolmogorov theorem
extension by conditioning
6. Martingales and Optional Times
96
21
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
7. Markov Processes and Discrete-Time Chains
117
19
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
8. Random Walks and Renewal Theory
136
20
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
9. Stationary Processes and Ergodic Theory
156
20
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
10. Poisson and Pure Jump-Type Markov Processes
176
23
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
11. Gaussian Processes and Brownian Motion
199
21
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-Ito integrals
chaos expansion of Brownian functionals
12. Skorohod Embedding and Invariance Principles
220
14
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
13. Independent Increments and Infinite Divisibility
234
21
regularity and jump structure
Levy representation
independent increments and infinite divisibility
stable processes
characteristics and convergence criteria
approximation of Levy processes and random walks
limit theorems for null arrays
convergence of extremes
14. Convergence of Random Processes, Measures, and Sets
255
20
relative compactness and tightness
uniform topology on C(K,S)
Skorohod's J(1)-topology
equicontinuity and tightness
convergence of random measures
superposition and thinning
exchangeable sequences and processes
simple point processes and random closed sets
15. Stochastic Integrals and Quadratic Variation
275
21
continuous local martingales and semimartingales
quadratic variation and covariation
existence and basic properties of the integral
integration by parts and Ito's formula
Fisk-Stratonovich integral
approximation and uniqueness
random time-change
dependence on parameter
16. Continuous Martingales and Brownian Motion
296
17
martingale characterization of Brownian motion
random time-change of martingales
isotropic local martingales
integral representation of martingales
interated and multiple integrals
change of measure and Girsanov's theorem
Cameron-Martin theorem
Wald's identity and Novikov's condition
17. Feller Processes and Semigroups
313
22
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
18. Stochastic Differential Equations and Martingale Problems
335
15
linear equations and Ornstein-Unlenbeck 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
transformations of SDEs
strong Markov and Feller properties
19. Local Time, Excursions, and Additive Functionals
350
21
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
20. One-Dimensional SDEs and Diffusions
371
19
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
21. PDE-Connections and Potential Theory
390
19
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
22. Predictability, Compensation, and Excessive Functions
409
24
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
23. Semimartingales and General Stochastic Integration
433
22
predictable convariation and L(2)-integral
semimartingale integral and covariation
general substitution rule
Doleans' exponential and change of measure
norm and exponential inequalities
martingale integral
decomposition of semimartingales
quasi-martingales and stochastic integrators
Appendices
455
9
A1. Hard Results in Measure Theory
A2. Some Special Spaces
Historical and Bibliographical Notes
464
22
Bibliography
486
23
Indices
509
Authors
Terms and Topics
Symbols