FOUNDATIONS OF MODERN PROBABILITY
CONTENTS 1 ELEMENTS OF MEASURE THEORY ?-?ELDS AND MONOTONE CLASSES MEASURABLE FUNCTIONS MEASURES AND INTEGRATION MONOTONE AND DOMINATED CONVERGENCE TRANSFORMATION OF INTEGRALS PRODUCT MEASURES AND FUBINI’S THEOREM LP-SPACES AND PROJECTION MEASURE SPACES AND KERNELS 2 PROCESSES, DISTRIBUTIONS, AND INDEPENDENCE 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 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 4 CHARACTERISTIC FUNCTIONS AND CLASSICAL LIMIT THEOREMS 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 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 ?LTRATIONS 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 MARKOV PROPERTY AND TRANSITION KERNELS ?NITE-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 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 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 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 SYMMETRIES OF GAUSSIAN DISTRIBUTION EXISTENCE AND PATH PROPERTIES OF BROWNIAN MOTION STRONG MARKOV AND RE?ECTION PROPERTIES ARCSINE AND UNIFORM LAWS LAW OF THE ITERATED LOGARITHM WIENER INTEGRALS AND ISONORMAL GAUSSIAN PROCESSES MULTIPLE WIENER–ITˆO INTEGRALS CHAOS EXPANSION OF BROWNIAN FUNCTIONALS 12 SKOROHOD EMBEDDING AND INVARIANCE PRINCIPLES 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 IN?NITE DIVISIBILITY REGULARITY AND JUMP STRUCTURE L´EVY REPRESENTATION INDEPENDENT INCREMENTS AND IN?NITE DIVISIBILITY STABLE PROCESSES CHARACTERISTICS AND CONVERGENCE CRITERIA APPROXIMATION OF L´EVY PROCESSES AND RANDOM WALKS LIMIT THEOREMS FOR NULL ARRAYS CONVERGENCE OF EXTREMES 14 CONVERGENCE OF RANDOM PROCESSES, MEASURES, AND SETS 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 15 STOCHASTIC INTEGRALS AND QUADRATIC VARIATION CONTINUOUS LOCAL MARTINGALES AND SEMIMARTINGALES QUADRATIC VARIATION AND COVARIATION EXISTENCE AND BASIC PROPERTIES OF THE INTEGRAL INTEGRATION BY PARTS AND ITˆO’S FORMULA FISK–STRATONOVICH INTEGRAL APPROXIMATION AND UNIQUENESS RANDOM TIME-CHANGE DEPENDENCE ON PARAMETER 16 CONTINUOUS MARTINGALES AND BROWNIAN MOTION 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 17 FELLER PROCESSES AND SEMIGROUPS SEMIGROUPS, RESOLVENTS, AND GENERATORS CLOSURE AND CORE HILLE–YOSIDA THEOREM EXISTENCE AND REGULARIZATION STRONG MARKOV PROPERTY CHARACTERISTIC OPERATOR DI?USIONS AND ELLIPTIC OPERATORS CONVERGENCE AND APPROXIMATION 18 STOCHASTIC DI?ERENTIAL EQUATIONS AND MARTINGALE PROBLEMS 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 TRANSFORMATIONS OF SDES STRONG MARKOV AND FELLER PROPERTIES 19 LOCAL TIME, EXCURSIONS, AND ADDITIVE FUNCTIONALS 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 DI?USIONS WEAK EXISTENCE AND UNIQUENESS PATHWISE UNIQUENESS AND COMPARISON SCALE FUNCTION AND SPEED MEASURE TIME-CHANGE REPRESENTATION BOUNDARY CLASSI?CATION ENTRANCE BOUNDARIES AND FELLER PROPERTIES RATIO ERGODIC THEOREM RECURRENCE AND ERGODICITY 21 PDE-CONNECTIONS AND POTENTIAL THEORY