THEORY OF PROBABILITY AND RANDOM PROCESSES

By RICHARD SANCHEZ

CONTENTS PART I PROBABILITY THEORY RANDOM VARIABLES AND THEIR DISTRIBUTIONS SPACES OF ELEMENTARY OUTCOMES, ?-ALGEBRAS, AND MEASURES EXPECTATION AND VARIANCE OF RANDOM VARIABLES ON A DISCRETE PROBABILITY SPACE PROBABILITY OF A UNION OF EVENTS EQUIVALENT FORMULATIONS OF ?-ADDITIVITY, BOREL ?-ALGEBRAS AND MEASURABILITY DISTRIBUTION FUNCTIONS AND DENSITIES PROBLEMS SEQUENCES OF INDEPENDENT TRIALS LAW OF LARGE NUMBERS AND APPLICATIONS DE MOIVRE-LAPLACE LIMIT THEOREM AND APPLICATIONS POISSON LIMIT THEOREM LEBESGUE INTEGRAL AND MATHEMATICAL EXPECTATION DE?NITION OF THE LEBESGUE INTEGRAL INDUCED MEASURES AND DISTRIBUTION FUNCTIONS TYPES OF MEASURES AND DISTRIBUTION FUNCTIONS REMARKS ON THE CONSTRUCTION OF THE LEBESGUE MEASURE CONVERGENCE OF FUNCTIONS, THEIR INTEGRALS, AND THE FUBINI THEOREM SIGNED MEASURES AND THE RADON-NIKODYM THEOREM LP SPACES MONTE CARLO METHOD CONDITIONAL PROBABILITIES AND INDEPENDENCE CONDITIONAL PROBABILITIES INDEPENDENCE OF EVENTS, ?-ALGEBRAS, AND RANDOM VARIABLES ?-SYSTEMS AND INDEPENDENCE MARKOV CHAINS WITH A FINITE NUMBER OF STATES STOCHASTIC MATRICES MARKOV CHAINS ERGODIC AND NON-ERGODIC MARKOV CHAINS LAW OF LARGE NUMBERS AND THE ENTROPY OF A MARKOV CHAIN PRODUCTS OF POSITIVE MATRICES GENERAL MARKOV CHAINS AND THE DOEBLIN CONDITION RANDOM WALKS ON THE LATTICE ZD RECURRENT AND TRANSIENT RANDOM WALKS RANDOM WALK ON Z AND THE RE?ECTION PRINCIPLE ARCSINE LAW GAMBLER’S RUIN PROBLEM LAWS OF LARGE NUMBERS DE?NITIONS, THE BOREL-CANTELLI LEMMAS, AND THE KOLMOGOROV INEQUALITY KOLMOGOROV THEOREMS ON THE STRONG LAW OF LARGE NUMBERS WEAK CONVERGENCE OF MEASURES DE?NITION OF WEAK CONVERGENCE WEAK CONVERGENCE AND DISTRIBUTION FUNCTIONS WEAK COMPACTNESS, TIGHTNESS, AND THE PROKHOROV THEOREM CHARACTERISTIC FUNCTIONS DE?NITION AND BASIC PROPERTIES CHARACTERISTIC FUNCTIONS AND WEAK CONVERGENCE GAUSSIAN RANDOM VECTORS LIMIT THEOREMS 10 1 CENTRAL LIMIT THEOREM, THE LINDEBERG CONDITION 10 2 LOCAL LIMIT THEOREM 10 3 CENTRAL LIMIT THEOREM AND RENORMALIZATION GROUP THEORY 10 4 PROBABILITIES OF LARGE DEVIATIONS 10 5 OTHER LIMIT THEOREMS 10 6 PROBLEMS SEVERAL INTERESTING PROBLEMS 11 1 WIGNER SEMICIRCLE LAW FOR SYMMETRIC RANDOM MATRICES 11 2 PRODUCTS OF RANDOM MATRICES 11 3 STATISTICS OF CONVEX POLYGONS PART II RANDOM PROCESSES BASIC CONCEPTS 12 1 DE?NITIONS OF A RANDOM PROCESS AND A RANDOM FIELD 12 2 KOLMOGOROV CONSISTENCY THEOREM 12 3 POISSON PROCESS 12 4 PROBLEMS CONDITIONAL EXPECTATIONS AND MARTINGALES 13 1 CONDITIONAL EXPECTATIONS 13 2 PROPERTIES OF CONDITIONAL EXPECTATIONS 13 3 REGULAR CONDITIONAL PROBABILITIES 13 4 FILTRATIONS, STOPPING TIMES, AND MARTINGALES 13 5 MARTINGALES WITH DISCRETE TIME 13 6 MARTINGALES WITH CONTINUOUS TIME 13 7 CONVERGENCE OF MARTINGALES 13 8 PROBLEMS MARKOV PROCESSES WITH A FINITE STATE SPACE 14 1 DE?NITION OF A MARKOV PROCESS 14 2 IN?NITESIMAL MATRIX 14 3 A CONSTRUCTION OF A MARKOV PROCESS 14 4 A PROBLEM IN QUEUING THEORY 14 5 PROBLEMS WIDE-SENSE STATIONARY RANDOM PROCESSES 15 1 HILBERT SPACE GENERATED BY A STATIONARY PROCESS 15 2 LAW OF LARGE NUMBERS FOR STATIONARY RANDOM PROCESSES 15 3 BOCHNER THEOREM AND OTHER USEFUL FACTS 15 4 SPECTRAL REPRESENTATION OF STATIONARY RANDOM PROCESSES 15 5 ORTHOGONAL RANDOM MEASURES 15 6 LINEAR PREDICTION OF STATIONARY RANDOM PROCESSES 15 7 STATIONARY RANDOM PROCESSES WITH CONTINUOUS TIME 15 8 PROBLEMS STRICTLY STATIONARY RANDOM PROCESSES 16 2 BIRKHO?ERGODIC THEOREM 16 3 ERGODICITY, MIXING, AND REGULARITY 16 4 STATIONARY PROCESSES WITH CONTINUOUS TIME 16 5 PROBLEMS GENERALIZED RANDOM PROCESSES 17 1 GENERALIZED FUNCTIONS AND GENERALIZED RANDOM PROCESSES 17 2 GAUSSIAN PROCESSES AND WHITE NOISE BROWNIAN MOTION 18 1 DE?NITION OF BROWNIAN MOTION 18 2 THE SPACE C([0, ?)) 18 3 EXISTENCE OF THE WIENER MEASURE, DONSKER THEOREM 18 4 KOLMOGOROV THEOREM 18 5 SOME PROPERTIES OF BROWNIAN MOTION 18 6 PROBLEMS MARKOV PROCESSES AND MARKOV FAMILIES 19 1 DISTRIBUTION OF THE MAXIMUM OF BROWNIAN MOTION 19 2 DE?NITION OF THE MARKOV PROPERTY 19 3 MARKOV PROPERTY OF BROWNIAN MOTION 19 4 THE AUGMENTED FILTRATION 19 5 DE?NITION OF THE STRONG MARKOV PROPERTY 19 6 STRONG MARKOV PROPERTY OF BROWNIAN MOTION 19 7 PROBLEMS STOCHASTIC INTEGRAL AND THE ITO FORMULA 20 1 QUADRATIC VARIATION OF SQUARE-INTEGRABLE MARTINGALES 20 2 THE SPACE OF INTEGRANDS FOR THE STOCHASTIC INTEGRAL 20 3 SIMPLE PROCESSES 20 4 DE?NITION AND BASIC PROPERTIES OF THE STOCHASTIC INTEGRAL 20 5 FURTHER PROPERTIES OF THE STOCHASTIC INTEGRAL 20 6 LOCAL MARTINGALES 20 7 ITO FORMULA 20 8 PROBLEMS STOCHASTIC DI?ERENTIAL EQUATIONS 21 1 EXISTENCE OF STRONG SOLUTIONS TO STOCHASTIC DI?ERENTIAL EQUATIONS 21 2 DIRICHLET PROBLEM FOR THE LAPLACE EQUATION 21 3 STOCHASTIC DI?ERENTIAL EQUATIONS AND PDE’S 21 4 MARKOV PROPERTY OF SOLUTIONS TO SDE’S 21 5 A PROBLEM IN HOMOGENIZATION 21 6 PROBLEMS 16 1 STATIONARY PROCESSES AND MEASURE PRESERVING TRANSFORMATIONS GIBBS RANDOM FIELDS 22 1 DE?NITION OF A GIBBS RANDOM FIELD 22 2 AN EXAMPLE OF A PHASE TRANSITION