PROBABILITY THEORY

By S. PAZAD

CONTENTS BASIC MEASURE THEORY CLASSES OF SETS SET FUNCTIONS THE MEASURE EXTENSION THEOREM MEASURABLE MAPS RANDOM VARIABLES INDEPENDENCE INDEPENDENCE OF EVENTS INDEPENDENT RANDOM VARIABLES KOLMOGOROV’S 0–1 LAW EXAMPLE: PERCOLATION GENERATING FUNCTIONS DE?NITION AND EXAMPLES POISSON APPROXIMATION BRANCHING PROCESSES THE INTEGRAL CONSTRUCTION AND SIMPLE PROPERTIES MONOTONE CONVERGENCE AND FATOU’S LEMMA LEBESGUE INTEGRAL VERSUS RIEMANN INTEGRAL MOMENTS AND LAWS OF LARGE NUMBERS MOMENTS WEAK LAW OF LARGE NUMBERS STRONG LAW OF LARGE NUMBERS SPEED OF CONVERGENCE IN THE STRONG LLN THE POISSON PROCESS CONVERGENCE THEOREMS ALMOST SURE AND MEASURE CONVERGENCE UNIFORM INTEGRABILITY EXCHANGING INTEGRAL AND DIFFERENTIATION LP-SPACES AND THE RADON–NIKODYM THEOREM DE?NITIONS INEQUALITIES AND THE FISCHER–RIESZ THEOREM HILBERT SPACES LEBESGUE’S DECOMPOSITION THEOREM SUPPLEMENT: SIGNED MEASURES SUPPLEMENT: DUAL SPACES CONDITIONAL EXPECTATIONS ELEMENTARY CONDITIONAL PROBABILITIES REGULAR CONDITIONAL DISTRIBUTION MARTINGALES PROCESSES, FILTRATIONS, STOPPING TIMES DISCRETE STOCHASTIC INTEGRAL DISCRETE MARTINGALE REPRESENTATION THEOREM AND THE CRR MODEL OPTIONAL SAMPLING THEOREMS DOOB DECOMPOSITION AND SQUARE VARIATION OPTIONAL SAMPLING AND OPTIONAL STOPPING UNIFORM INTEGRABILITY AND OPTIONAL SAMPLING MARTINGALE CONVERGENCE THEOREMS AND THEIR APPLICATIONS DOOB’S INEQUALITY MARTINGALE CONVERGENCE THEOREMS EXAMPLE: BRANCHING PROCESS BACKWARDS MARTINGALES AND EXCHANGEABILITY EXCHANGEABLE FAMILIES OF RANDOM VARIABLES BACKWARDS MARTINGALES DE FINETTI’S THEOREM CONVERGENCE OF MEASURES A TOPOLOGY PRIMER WEAK AND VAGUE CONVERGENCE PROHOROV’S THEOREM APPLICATION: A FRESH LOOK AT DE FINETTI’S THEOREM PROBABILITY MEASURES ON PRODUCT SPACES PRODUCT SPACES FINITE PRODUCTS AND TRANSITION KERNELS KOLMOGOROV’S EXTENSION THEOREM MARKOV SEMIGROUPS CHARACTERISTIC FUNCTIONS AND THE CENTRAL LIMIT THEOREM SEPARATING CLASSES OF FUNCTIONS CHARACTERISTIC FUNCTIONS: EXAMPLES LÉVY’S CONTINUITY THEOREM CHARACTERISTIC FUNCTIONS AND MOMENTS THE CENTRAL LIMIT THEOREM MULTIDIMENSIONAL CENTRAL LIMIT THEOREM IN?NITELY DIVISIBLE DISTRIBUTIONS LÉVY–KHINCHIN FORMULA STABLE DISTRIBUTIONS MARKOV CHAINS DE?NITIONS AND CONSTRUCTION DISCRETE MARKOV CHAINS: EXAMPLES DISCRETE MARKOV PROCESSES IN CONTINUOUS TIME DISCRETE MARKOV CHAINS: RECURRENCE AND TRANSIENCE APPLICATION: RECURRENCE AND TRANSIENCE OF RANDOM WALKS INVARIANT DISTRIBUTIONS STOCHASTIC ORDERING AND COUPLING CONVERGENCE OF MARKOV CHAINS PERIODICITY OF MARKOV CHAINS COUPLING AND CONVERGENCE THEOREM MARKOV CHAIN MONTE CARLO METHOD SPEED OF CONVERGENCE MARKOV CHAINS AND ELECTRICAL NETWORKS HARMONIC FUNCTIONS REVERSIBLE MARKOV CHAINS FINITE ELECTRICAL NETWORKS RECURRENCE AND TRANSIENCE NETWORK REDUCTION RANDOM WALK IN A RANDOM ENVIRONMENT ERGODIC THEORY ERGODIC THEOREMS EXAMPLES APPLICATION: RECURRENCE OF RANDOM WALKS MIXING ENTROPY