Chapter
1.9 Extremal solutions of (general) moments problems
1.10 The log normal distribution is moments indeterminate
1.11 Conditional expectations and equality in law
1.12 Simplifiable random variables .
1.13 Mellin transform and simplification
Solutions for Chapter 1 .
2 Independence and conditioning
2.1 Independence does not imply measurability with respect to an inde-pendent complement
2.2 Complement to Exercise 2.1: further statements of independence versus measurability
2.3 Independence and mutual absolute continuity
2.4 Size-biased sampling and conditional laws
2.5 Think twice before exchanging the order of taking the supremum and intersection of σ-fields!
2.6 Exchangeability and conditional independence: de Finetti’s theorem
2.7 Too much independence implies constancy
2.8 A double paradoxical inequality
2.9 Euler’s formula for primes and probability
2.10 The probability, for integers, of being relatively prime
2.11 Bernoulli random walks considered at some stopping time
2.12 cosh, sinh, the Fourier transform and conditional independence
2.13 cosh, sinh, and the Laplace transform
2.14 Conditioning and changes of probabilities
2.15 Radon–Nikodym density and the Acceptance–Rejection Method of von Neumann
2.16 Negligible sets and conditioning
2.17 Gamma laws and conditioning
2.18 Random variables with independent fractional and integer parts
Solution to Exercise 2.6:
Solution to Exercise 2.10
Solution to Exercise 2.11
Solution to Exercise 2.12
Solution to Exercise 2.13
Solution to Exercise 2.14
Solution to Exercise 2.15
Solution to Exercise 2.16
Solution to Exercise 2.17
Solution to Exercise 2.18
3.1 Constructing Gaussian variables from, but not belonging to, a Gaus-sian space
3.2 A complement to Exercise 3.1
3.3 On the negative moments of norms of Gaussian vectors
3.4 Quadratic functionals of Gaussian vectors and continued fractions
3.5 Orthogonal but non-independent Gaussian variables
3.6 Isotropy property of multidimensional Gaussian laws
3.7 The Gaussian distribution and matrix transposition
3.8 A law whose n-samples are preserved by every orthogonal transfor-mation is Gaussian
3.9 Non-canonical representation of Gaussian random walks
3.10 Concentration inequality for Gaussian vectors
3.11 Determining a jointly Gaussian distribution from its conditional marginals
Solution to Exercise 3.10
Solution to Exercise 3.11
4 Distributional computations
4.1 Hermite polynomials and Gaussian variables
4.2 The beta–gamma algebra and Poincar´e’s Lemma
4.3 An identity in law between reciprocals of gamma variables
4.4 The Gamma process and its associated Dirichlet processes.
4.5 Gamma variables and Gauss multiplication formulae
4.6 The beta–gamma algebra and convergence in law
4.7 Beta–gamma variables and changes of probability measures
4.8 Exponential variables and powers of Gaussian variables
4.9 Mixtures of exponential distributions
4.10 Some computations related to the lack of memory property of the exponential law
4.11 Some identities in law between Gaussian and exponential variables
4.12 Some functions which preserve the Cauchy law
4.13 Uniform laws on the circle
4.14 Trigonometric formulae and probability
4.15 A multidimensional version of the Cauchy distribution
4.16 Some properties of the Gauss transform
4.17 Unilateral stable distributions (1)
4.18 Unilateral stable distributions (2)
4.19 Unilateral stable distributions (3)
4.20 A probabilistic translation of Selberg’s integral formulae
4.21 Mellin and Stieltjes transforms of stable variables
4.22 Solving certain moment problems via simplification
Solution to Exercise 4.10
Solution to Exercise 4.11
Solution to Exercise 4.12
Solution to Exercise 4.13
Solution to Exercise 4.14
Solution to Exercise 4.15
Solution to Exercise 4.16
Solution to Exercise 4.17
Solution to Exercise 4.18
Solution to Exercise 4.19
Solution to Exercise 4.20
Solution to Exercise 4.21
Solution to Exercise 4.22
5 Convergence of random variables
5.1 Convergence of sum of squares of independent Gaussian variables
5.2 Convergence of moments and convergence in law
5.3 Borel test functions and convergence in law
5.4 Convergence in law of the normalized maximum of Cauchy variables
5.5 Large deviations for the maximum of Gaussian vectors
5.6 A logarithmic normalization
5.7 A vn log n normalization
5.8 The Central Limit Theorem involves convergence in law, not in prob-ability
5.9 Changes of probabilities and the Central Limit Theorem
5.10 Convergence in law of stable(µ) variables, as µ . 0
5.11 Finite dimensional convergence in law towards Brownian motion
5.12 The empirical process and the Brownian bridge
5.13 The Poisson process and Brownian motion
5.14 Brownian bridges converging in law to Brownian motions.
Solution to Exercise 5.10
Solution to Exercise 5.11
Solution to Exercise 5.12
Solution to Exercise 5.13
Solution to Exercise 5.14
Solution to Exercise 5.15
6.1 Solving a particular SDE .
6.2 The range process of Brownian motion
6.3 Symmetric Levy processes reflected at their minimum and maximum; E. Cs´aki’s formulae for the ratio of Brownian extremes
6.4 A toy example for Westwater’s renormalization
6.5 Some asymptotic laws of planar Brownian motion
6.6 Windings of the three-dimensional Brownian motion around a line . .
6.7 Cyclic exchangeability property and uniform law related to the Brownian bridge
6.8 Local time and hitting time distributions for the Brownian bridge
6.9 Partial absolute continuity of the Brownian bridge distribution with respect to the Brownian distribution
6.10 A Brownian interpretation of the duplication formula for the gamma function
6.11 Some deterministic time-changes of Brownian motion
6.12 Random scaling of the Brownian bridge
6.13 Time-inversion and quadratic functionals of Brownian motion; L´evy’s stochastic area formula
6.14 Quadratic variation and local time of semimartingales
6.15 Geometric Brownian motion
6.16 0-self similar processes and conditional expectation
6.17 A Taylor formula for semimartingales; Markov martingales and iter-ated infinitesimal generators
6.18 A remark of D. Williams: the optional stopping theorem may hold for certain “non-stopping times”
6.19 Stochastic affine processes, also known as “Harnesses”.
6.20 A martingale “in the mean over time” is a martingale
6.21 A reinforcement of Exercise 6.20
Solution to Exercise 6.10
Solution to Exercise 6.11
Solution to Exercise 6.12
Solution to Exercise 6.13
Solution to Exercise 6.14
Solution to Exercise 6.15
Solution to Exercise 6.16
Solution to Exercise 6.17
Solution to Exercise 6.19
Joint solution to Exercises 6.20 and 6.21
Where is the notion N discussed ?
Final suggestions: how to go further ?
Exercises in Probability
：A Guided Tour from Measure Theory to Random Processes, via Conditioning
( Cambridge Series in Statistical and Probabilistic Mathematics )
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