10 CONTENTS

27.1 The Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . 73527.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 73627.3 Conditional Expectation, Sub-martingales . . . . . . . . . . . . . . . . . 74027.4 Characteristic Functions and Independence . . . . . . . . . . . . . . . . . 74427.5 Characteristic Functions for Measures . . . . . . . . . . . . . . . . . . . 74827.6 Independence in Banach Space . . . . . . . . . . . . . . . . . . . . . . . 75027.7 Convolution and Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752

28 The Normal Distribution 75928.1 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . 75928.2 Linear Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76228.3 Finding Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76528.4 Prokhorov and Levy Theorems . . . . . . . . . . . . . . . . . . . . . . . 76628.5 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 776

29 Martingales 78129.1 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . 78129.2 Conditional Expectation and Independence . . . . . . . . . . . . . . . . . 78429.3 Discrete Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . 786

29.3.1 Upcrossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78829.3.2 The Sub-martingale Convergence Theorem . . . . . . . . . . . . 79029.3.3 Doob Sub-martingale Estimates . . . . . . . . . . . . . . . . . . 793

29.4 Optional Sampling and Stopping Times . . . . . . . . . . . . . . . . . . . 79529.4.1 Optional Sampling for Martingales . . . . . . . . . . . . . . . . 79829.4.2 Optional Sampling Theorem for Sub-Martingales . . . . . . . . 799

29.5 Reverse Sub-martingale Convergence Theorem . . . . . . . . . . . . . . . 80229.6 Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . 804

30 Continuous Stochastic Processes 80730.1 Fundamental Definitions and Properties . . . . . . . . . . . . . . . . . . . 80730.2 Kolmogorov Čentsov Continuity Theorem . . . . . . . . . . . . . . . . . 80930.3 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81730.4 Martingales and Sub-Martingales . . . . . . . . . . . . . . . . . . . . . . 82530.5 Some Maximal Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 826

31 Optional Sampling Theorems 83131.1 Review of Discreet Stopping Times . . . . . . . . . . . . . . . . . . . . . 83131.2 Review of Doob Optional Sampling Theorem . . . . . . . . . . . . . . . 83331.3 Doob Optional Sampling Continuous Case . . . . . . . . . . . . . . . . . 834

31.3.1 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . 83431.3.2 The Optional Sampling Theorem Continuous Case . . . . . . . . 839

31.4 Maximal Inequalities and Stopping Times . . . . . . . . . . . . . . . . . 84531.5 Continuous Sub-martingale Convergence . . . . . . . . . . . . . . . . . . 84931.6 Hitting This Before That . . . . . . . . . . . . . . . . . . . . . . . . . . . 85331.7 The Space M p

T (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856

32 Quadratic Variation 85932.1 How to Recognize a Martingale . . . . . . . . . . . . . . . . . . . . . . . 85932.2 Martingales and Total Variation . . . . . . . . . . . . . . . . . . . . . . . 862