Contents Preface to the Second Edition v Preface to the First Edition vii 1. Measure Theory-Basic Notions 1 Measurable sets and functions measures and integration monotone and dominated convergence transformation of integrals product measures and Fubinis theorem LP-spaces and projection approximation measure spaces and kernels 2. Measure Theory——Key Results 23 Outer measures and extension Lebesgue and Lebesgue-Stieltjes measures Jordan-llahn and Lebesgue decompositions Radon-Nikodym theorem Lebesgues differentiation theorem functions of jinite varvation Riesz representation theorem Haar and invariant measures 3. Processes,Distributions,and Independence 45 Random elements and processes distributions and erpectatiorz independence zero-one laws Borel-Cantelli lemma Bernoulli seqzrences and existence moments and continuity of paths 4. Random Sequences,Series,and Averages 62 Convergence in probability and in Lp 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 5. Characteristic Functions and Classical Limit Theorems 83 Uniqueness afnd continuity theoretm Poisson convergence positive and symmetric terms Lindebergs condition general Gaussiatn conuergence weak laws of large numbers docmain of Gaussian attraction vague arnd weak compactness 6. Conditioning and Disintegration 103 Conditiotnal expectations and probabilities regular conditiotnal distributions disintegration conditional independence transfer and coupling existence of sequences and processes extension through conditioning 7. Martingales and Optional Times 119 Filtrrations atnd optional times random time-change martingale property optional stopping and sampling tmaxitmum and upcrossing inequalities martingale convergence regularity and closure limits of conditional expectations regularization of submartingales 8. Markov Processes and Discrete-Time Chains 140 Matrkov property and transition kernels finite-dimensional distributions atnd existence space and time homogeneity strong Mafrkou property arnd excursions invariant distributions and stationarity recurrence and transience ergodic behavior of irreducible chains mean recurrence times 9. Random Walks and Renewal Theory 159 Recurrecnce and transience dependence on dimensiotn general recurrence criteria symmetry and duality Wiener-Hopf factorization ladder time and height distn:bution stationary renewal process renewal theorem 10- Stationary Processes and Ergodic Theory 178 Stationanty invariance and ergodicity discrete-aud continuous-time ergodic theorems moment and maximum inequalities multivariate ergodic theorems sample intensity of a random measure subadditivity and products of random matrices conditioning ancl ergodic decomposition shzlt coupling and the invariant a-field 11. Special Notions of Symmetry and Invariance 202 Palm distn:butions atnd inversiorn formulas stationarity and cycle stationarity local hitting and conditioning ergodic properties of Palm measures exchangeable sequences and processes strong stationarity and predictable sampling ballot theorems entropy and information 12. Poisson and Pure Jump-Type Markov Processes 224 Random measures and point processes Cox processes randomization and thinning mixed Poisson and binomial processes independence and symmetry critert:a Markov transition and rate kernels embedded Markov chains and erplosion compound and pseudo-Poisson processes ergodic behavior of irreducible chains 13. Gaussian Processes and Brownian Motion 249 Symmetries of Gaussian distributiotn existence and path properties of Btrownian motion strong Markou and reflection properties arcsine atnd uniform laws law of the iterated logarithm Wiener integrals and isonormal Gaussian processes multiple Wiener-Ito integrals chaos erpansion of Brownian functionals 14. Skorohod Embedding and Invariance Principles 270 Embedding of random variables approximation of random walks functional central limit theorerm laws of the iterated logarithm arcsine laws approximation of rrefnewal processes empirical distribution jurtctions embedding afnd approximation of martingales 15. Independent Increments and Infinite Divisibility 285 Regularity and integral representation Levy proces8es and subordinators stable processes and Jirst-passage times infinitely divisible distributions chatracterristics and convergence criteria approximation of Levy processes and ratndom walks limit theorems for null arrays convergence of extremes 16. Convergence of Random Processes Measures and Sets 307 Relative compactness and tightness uniform topology on C(K,S) Skorohods Ji-topology equicontinuity atnd tightness convergence of random measures superposition and thinning exchangeable sequences alnd processes simple point processes and random closed sets 17. Stochastic Integrals and Quadratic Variation 329 Continuous local martingales and semimartingales quadratic valriation and covariation existence atnd basic properties of the integral integration by parts and Itos formula Fisk-Stratonovich integral approximation and uniqueness random time-change dependence on parameter 18. Continuous Martingales and Brownian Motion 350 Real and complex exponentiat martingales martingale characterization of Brownian motion trandom time-change of martingales integral representation of martingales iterated and multiple integrals change of measure and Girsanovs theorem Cameron-Martin theorem Walds identity and Novikovs condition 19. Feller Processes and Semigroups 367 Semigroups resolvents and generators closure and core Hille-Yosida theorem eristence and regularization strong Markov property characteristic operator dibcusions and elliptic operutors conuergence and approrLmation 20. Ergodic Properties of Markov Processes 390 transition and contraction operators ratio ergodic theorem space-time invan:ance and tail trtviality mixmg and convergence in total variation Harris recurrence and transience eristence and uniqueness of invariarzt measure distrzbutional and pathwise limits 21. Stochastic Differential Equations and Martingale Problems 412 Linear equations and Ornstem-Uhlenbeck processes strong existence uniqueness and nonextplosion criten:a weak solutiotns and local martingale problems well-posedness and measurrability pathwise uniqueness and functional solution weak existence and continrrity transformation of SDEs strong Markov and Feller properties 22. Local Time Excursions and Additive Functionals 428 Tanakas formula and semimartingale local time occupation density continuity and approrimation regenerative sets and processes excursion local time and Poisson process Ray-Knight theorem excessive functions and additive functionals local time at a regrrlar point additive functionals of Brownian motion 23. One-dimensional SDEs and Diffusions 450 Weak existence and uniqueness pathwise uniqueness and comparison scale junction and speed measure time-change representation boundary class~cation entrance boundaries atnd Feller properties ratio ergodic theorem recurrence and ergodicity 24. Connections with PDEs and Potential Theory 470 Backward equation atnd Feynman-Kac formula uniqueness for SDEs from existence for PDEs harmonic functions and Dirichlets problem Green functions as occupation densities sweeping and equilibrium problems dependence on conductor and domam time reversal capacities and randotm sets 25. Predictability Compensation and Excessive Functions 490 Accessible and predictable times natural and predictable processes Doob-Meyer decomposition quasi-left-continuity compensation of randocm measures excessive and superharmonic junctions additive functionals as compensators Riesz decomposition 26. Semimartingales and General Stochastic Integration 515 Predictable covafriation and L2-itntegral semimartingale integral and covariation general szrbstitution rule Doleans exponential and change of measure norm and exponential inequalities martingale integral decomposition of semimartingales quasi martingales and stochastic integrators 27. Large Deviations 537 Legeudre-Fenchel transform Cramers and Schilders theorems large-deviation principle and rate function functional form of the LDP continuous mapping and extension perturbation of dynamical systems empincal processes and entropy Strassens law of the iterated logarithm Appendices A1. Advanced Measure Theory 561 Polish and Borel spaces measurable inverses projection and sections A2. Some Special Spaces 562 Function spaces measure spaces spaces of closed sets measure-valued junctions projective limits Historical and Bibliographical Notes 569 Bibliography 596 Symbol Index 621 Author Index 623 Subject Index 629
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