16 CONTENTS

58.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1851

VI Topics In Probability 1853

59 Basic Probability 185559.1 Random Variables And Independence . . . . . . . . . . . . . . . . . . . .185559.2 Kolmogorov Extension Theorem For Polish Spaces . . . . . . . . . . . .186159.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .186459.4 Banach Space Valued Random Variables . . . . . . . . . . . . . . . . . .186959.5 Reduction To Finite Dimensions . . . . . . . . . . . . . . . . . . . . . .187259.6 0,1 Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187359.7 Kolmogorov’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . .187659.8 The Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . .188259.9 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . .188359.10 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . .188759.11 Characteristic Functions, Independence . . . . . . . . . . . . . . . . . . .189159.12 Characteristic Functions For Measures . . . . . . . . . . . . . . . . . . .189559.13 Characteristic Functions In Banach Space . . . . . . . . . . . . . . . . .189859.14 Convolution And Sums . . . . . . . . . . . . . . . . . . . . . . . . . . .190059.15 The Convergence Of Sums . . . . . . . . . . . . . . . . . . . . . . . . .190559.16 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . .191059.17 Use Of Characteristic Functions To Find Moments . . . . . . . . . . . . .191759.18 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . .191959.19 Characteristic Functions, Prokhorov Theorem . . . . . . . . . . . . . . .192659.20 Generalized Multivariate Normal . . . . . . . . . . . . . . . . . . . . . .193259.21 Positive Definite Functions, Bochner’s Theorem . . . . . . . . . . . . . .1937

60 Conditional, Martingales 194560.1 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . .194560.2 Discrete Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1948

60.2.1 Upcrossings . . . . . . . . . . . . . . . . . . . . . . . . . . . .195060.2.2 The Submartingale Convergence Theorem . . . . . . . . . . . .195260.2.3 Doob Submartingale Estimate . . . . . . . . . . . . . . . . . . .1953

60.3 Optional Sampling And Stopping Times . . . . . . . . . . . . . . . . . .195360.3.1 Stopping Times And Their Properties Overview . . . . . . . . .1953

60.4 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195660.5 Optional Stopping Times And Martingales . . . . . . . . . . . . . . . . .1960

60.5.1 Stopping Times And Their Properties . . . . . . . . . . . . . . .196060.6 Submartingale Convergence Theorem . . . . . . . . . . . . . . . . . . . .1966

60.6.1 Upcrossings . . . . . . . . . . . . . . . . . . . . . . . . . . . .196660.6.2 Maximal Inequalities . . . . . . . . . . . . . . . . . . . . . . .196960.6.3 The Upcrossing Estimate . . . . . . . . . . . . . . . . . . . . .1971

60.7 The Submartingale Convergence Theorem . . . . . . . . . . . . . . . . .197460.8 A Reverse Submartingale Convergence Theorem . . . . . . . . . . . . . .1978