PROBABILITY, RANDOM PROCESSES, AND ERGODIC PROPERTIES

By BARBARA LEWIS

CONTENTS PROBABILITY AND RANDOM PROCESSES INTRODUCTION PROBABILITY SPACES AND RANDOM VARIABLES RANDOM PROCESSES AND DYNAMICAL SYSTEMS DISTRIBUTIONS EXTENSION ISOMORPHISM STANDARD ALPHABETS EXTENSION OF PROBABILITY MEASURES STANDARD SPACES SOME PROPERTIES OF STANDARD SPACES SIMPLE STANDARD SPACES METRIC SPACES EXTENSION IN STANDARD SPACES THE KOLMOGOROV EXTENSION THEOREM EXTENSION WITHOUT A BASIS BOREL SPACES AND POLISH ALPHABETS BOREL SPACES POLISH SPACES POLISH SCHEMES AVERAGES DISCRETE MEASUREMENTS QUANTIZATION EXPECTATION TIME AVERAGES CONVERGENCE OF RANDOM VARIABLES STATIONARY AVERAGES CONDITIONAL PROBABILITY AND EXPECTATION MEASUREMENTS AND EVENTS RESTRICTIONS OF MEASURES ELEMENTARY CONDITIONAL PROBABILITY PROJECTIONS THE RADON-NIKODYM THEOREM CONDITIONAL PROBABILITY REGULAR CONDITIONAL PROBABILITY CONDITIONAL EXPECTATION 5 10 INDEPENDENCE AND MARKOV CHAINS ERGODIC PROPERTIES ERGODIC PROPERTIES OF DYNAMICAL SYSTEMS SOME IMPLICATIONS OF ERGODIC PROPERTIES ASYMPTOTICALLY MEAN STATIONARY PROCESSES RECURRENCE ASYMPTOTIC MEAN EXPECTATIONS LIMITING SAMPLE AVERAGES ERGODICITY ERGODIC THEOREMS THE POINTWISE ERGODIC THEOREM BLOCK AMS PROCESSES THE ERGODIC DECOMPOSITION THE SUBADDITIVE ERGODIC THEOREM PROCESS METRICS AND THE ERGODIC DECOMPOSITION A METRIC SPACE OF MEASURES THE RHO-BAR DISTANCE MEASURES ON MEASURES THE ERGODIC DECOMPOSITION REVISITED THE ERGODIC DECOMPOSITION OF MARKOV PROCESSES BARYCENTERS A?NE FUNCTIONS OF MEASURES THE ERGODIC DECOMPOSITION OF A?NE FUNCTIONALS