ADVANCED PROBABILITY University of Cambridge, Part III of the Mathematical Tripos Michaelmas Term 2006 Grégory Miermont1 1CNRS & Laboratoire de Mathématique, Equipe Probabilités, Statistique et Modélisation, Bt. 425, Université Paris-Sud, 91405 Orsay, France
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Contents 1 Conditional expectation 5 1.1 The discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Conditioning with respect to a o-algebra 6 1.2.1 The L2 case 7 1.2.2 General case 7 1.2.3 Non-negative case. 8 1.3 Specific properties of conditional expectation 10 1.4 Computing a conditional expectation 1.4.1 Conditional density functions 11 11 1.4.2 The Gaussian case . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Discrete-time martingales 13 2.1.1 Stochastic processes, filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Basic notions . . . . . . . . 13 2.1.2 Martingales 14 2.1.3 Doob's stopping times 14 2.2 Optional stopping 15 2.3 The convergence theorem . 17 2.4 LP convergence, p > 1 . 18 2.4.1 A maximal inequality 18 2.5 L1 convergence 20 2.6 Optional stopping in the UI case 2.7 Backwards martingales 21 . . . . . . . . . . . . . . . . . . . . . . . 20 3 Examples of applications of discrete-time martingales 23 3.1 Kolmogorov's 0 - 1 law, law of large numbers 23 3.2 Branching processes . 23 3.3 The Radon-Nikodym theorem 24 3.4 Product martingales 26 3.4.1 Example: consistency of the likelihood ratio test 28 3 -----