Exercises in probability

Cover......Page 1
About......Page 2
CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS......Page 3
Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning......Page 4
Copyright - ISBN: 0521825857......Page 5
Contents......Page 8
Preface......Page 14
Some frequently used notations......Page 16
1.1 Sets which do not belong in a strong sense, to a σ-field......Page 18
1.2 Some criteria for uniform integrability......Page 20
1.3 When does weak convergence imply the convergence of expectations?......Page 21
1.5 L^p-convergence of conditional expectations......Page 22
1.7 Ergodic transformations......Page 23
1.8 Invariant σ-fields......Page 24
1.9 Extremal solutions of (general) moments problems......Page 25
1.10 The log normal distribution is moments indeterminate......Page 26
1.11 Conditional expectations and equality in law......Page 27
1.12 Simplifiable random variables......Page 28
1.13 Mellin transform and simplification......Page 29
Solutions for Chapter 1......Page 30
2 Independence and conditioning......Page 42
2.1 Independence does not imply measurability with respect to an independent complement......Page 43
2.3 Independence and mutual absolute continuity......Page 44
2.4 Size-biased sampling and conditional laws......Page 45
2.5 Think twice before exchanging the order of taking the supremum and intersection of σ-fields!......Page 46
2.6 Exchangeability and conditional independence: de Finetti’s theorem......Page 47
2.7 Too much independence implies constancy......Page 48
2.8 A double paradoxical inequality......Page 49
2.9 Euler’s formula for primes and probability......Page 50
2.10 The probability, for integers, of being relatively prime......Page 51
2.11 Bernoulli random walks considered at some stopping time......Page 52
2.12 cosh, sinh, the Fourier transform and conditional independence......Page 53
2.13 cosh, sinh, and the Laplace transform......Page 54
2.14 Conditioning and changes of probabilities......Page 55
2.16 Negligible sets and conditioning......Page 56
2.17 Gamma laws and conditioning......Page 58
2.18 Random variables with independent fractional and integer parts......Page 59
Solutions for Chapter 2......Page 60
3 Gaussian variables......Page 84
3.2 A complement to Exercise 3.1......Page 85
3.3 On the negative moments of norms of Gaussian vectors......Page 86
3.4 Quadratic functionals of Gaussian vectors and continued fractions......Page 87
3.5 Orthogonal but non-independent Gaussian variables......Page 89
3.7 The Gaussian distribution and matrix transposition......Page 90
3.9 Non-canonical representation of Gaussian random walks......Page 91
3.10 Concentration inequality for Gaussian vectors......Page 93
3.11 Determining a jointly Gaussian distribution from its conditional marginals......Page 94
Solutions for Chapter 3......Page 95
4 Distributional computations......Page 108
4.1 Hermite polynomials and Gaussian variables......Page 109
4.2 The beta–gamma algebra and Poincaré’s Lemma......Page 110
4.3 An identity in law between reciprocals of gamma variables......Page 113
4.4 The Gamma process and its associated Dirichlet processes......Page 114
4.5 Gamma variables and Gauss multiplication formulae......Page 115
4.7 Beta–gamma variables and changes of probability measures......Page 117
4.8 Exponential variables and powers of Gaussian variables......Page 118
4.9 Mixtures of exponential distributions......Page 119
4.10 Some computations related to the lack of memory property of the exponential law......Page 120
4.11 Some identities in law between Gaussian and exponential variables......Page 121
4.13 Uniform laws on the circle......Page 122
4.15 A multidimensional version of the Cauchy distribution......Page 123
4.16 Some properties of the Gauss transform......Page 125
4.17 Unilateral stable distributions (1)......Page 127
4.18 Unilateral stable distributions (2)......Page 128
4.19 Unilateral stable distributions (3)......Page 129
4.20 A probabilistic translation of Selberg’s integral formulae......Page 132
4.21 Mellin and Stieltjes transforms of stable variables......Page 133
4.22 Solving certain moment problems via simplification......Page 134
Solutions for Chapter 4......Page 136
5 Convergence of random variables......Page 166
5.3 Borel test functions and convergence in law......Page 167
5.5 Large deviations for the maximum of Gaussian vectors......Page 168
5.6 A logarithmic normalization......Page 169
5.8 The Central Limit Theorem involves convergence in law, not in probability......Page 170
5.9 Changes of probabilities and the Central Limit Theorem......Page 171
5.11 Finite dimensional convergence in law towards Brownian motion......Page 172
5.13 The Poisson process and Brownian motion......Page 174
5.14 Brownian bridges converging in law to Brownian motions......Page 175
5.15 An almost sure convergence result for sums of stable random variables......Page 176
Solutions for Chapter 5......Page 178
6 Random processes......Page 192
6.1 Solving a particular SDE......Page 194
6.3 Symmetric Lévy processes reflected at their minimum and maximum; E. Csáki’s formulae for the ratio of Brownian extremes......Page 195
6.4 A toy example for Westwater’s renormalization......Page 197
6.5 Some asymptotic laws of planar Brownian motion......Page 199
6.6 Windings of the three-dimensional Brownian motion around a line......Page 200
6.7 Cyclic exchangeability property and uniform law related to the Brownian bridge......Page 201
6.8 Local time and hitting time distributions for the Brownian bridge......Page 202
6.9 Partial absolute continuity of the Brownian bridge distribution with respect to the Brownian distribution......Page 204
6.10 A Brownian interpretation of the duplication formula for the gamma function......Page 205
6.11 Some deterministic time-changes of Brownian motion......Page 206
6.12 Random scaling of the Brownian bridge......Page 207
6.13 Time-inversion and quadratic functionals of Brownian motion; L´evy’s stochastic area formula......Page 208
6.15 Geometric Brownian motion......Page 210
6.16 0-self similar processes and conditional expectation......Page 212
6.17 A Taylor formula for semimartingales; Markov martingales and iterated infinitesimal generators......Page 213
6.18 A remark of D. Williams: the optional stopping theorem may hold for certain “non-stopping times”......Page 214
6.19 Stochastic affine processes, also known as “Harnesses”......Page 215
6.20 A martingale “in the mean over time” is a martingale......Page 217
6.21 A reinforcement of Exercise 6.20......Page 218
Solutions for Chapter 6......Page 219
Where is the notion N discussed ?......Page 243
Final suggestions: how to go further ?......Page 244
References......Page 246
Index......Page 252