CONTENTS 17

60.9 Strong Law Of Large Numbers . . . . . . . . . . . . . . . . . . . . . . .1980

61 Probability In Infinite Dimensions 198561.1 Conditional Expectation In Banach Spaces . . . . . . . . . . . . . . . . .198561.2 Probability Measures And Tightness . . . . . . . . . . . . . . . . . . . .198861.3 Tight Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199061.4 A Major Existence And Convergence Theorem . . . . . . . . . . . . . . .199561.5 Bochner’s Theorem In Infinite Dimensions . . . . . . . . . . . . . . . . .200261.6 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . .200661.7 Gaussian Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2011

61.7.1 Definitions And Basic Properties . . . . . . . . . . . . . . . . .201161.7.2 Fernique’s Theorem . . . . . . . . . . . . . . . . . . . . . . . .2015

61.8 Gaussian Measures For A Separable Hilbert Space . . . . . . . . . . . . .202061.9 Abstract Wiener Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .202961.10 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204361.11 Existence Of Abstract Wiener Spaces . . . . . . . . . . . . . . . . . . . .2043

62 Stochastic Processes 204962.1 Fundamental Definitions And Properties . . . . . . . . . . . . . . . . . .204962.2 Kolmogorov Čentsov Continuity Theorem . . . . . . . . . . . . . . . . .205162.3 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206062.4 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206862.5 Some Maximal Estimates . . . . . . . . . . . . . . . . . . . . . . . . . .206962.6 Optional Sampling Theorems . . . . . . . . . . . . . . . . . . . . . . . .2072

62.6.1 Stopping Times And Their Properties . . . . . . . . . . . . . . .207262.6.2 Doob Optional Sampling Theorem . . . . . . . . . . . . . . . .2076

62.7 Doob Optional Sampling Continuous Case . . . . . . . . . . . . . . . . .208062.7.1 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . .208062.7.2 The Optional Sampling Theorem Continuous Case . . . . . . . .2085

62.8 Right Continuity Of Submartingales . . . . . . . . . . . . . . . . . . . .209162.9 Some Maximal Inequalities . . . . . . . . . . . . . . . . . . . . . . . . .209762.10 Continuous Submartingale Convergence Theorem . . . . . . . . . . . . .210162.11 Hitting This Before That . . . . . . . . . . . . . . . . . . . . . . . . . . .210662.12 The Space M p

T (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2110

63 The Quadratic Variation Of A Martingale 211563.1 How To Recognize A Martingale . . . . . . . . . . . . . . . . . . . . . .211563.2 The Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . .211963.3 The Covariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212763.4 The Burkholder Davis Gundy Inequality . . . . . . . . . . . . . . . . . .213063.5 The Quadratic Variation And Stochastic Integration . . . . . . . . . . . .213763.6 Another Limit For Quadratic Variation . . . . . . . . . . . . . . . . . . .214463.7 Doob Meyer Decomposition . . . . . . . . . . . . . . . . . . . . . . . . .215063.8 Levy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2169

64 Wiener Processes 2179