Actions for Exercises in probability : a guided tour from measure theory to random processes, via conditioning
Exercises in probability : a guided tour from measure theory to random processes, via conditioning / Loïc Chaumont, Marc Yor
- Author
- Chaumont, L. (Loïc)
- Published
- Cambridge : Cambridge University Press, 2012.
- Edition
- 2nd ed.
- Physical Description
- 1 online resource (xx, 279 pages).
- Additional Creators
- Yor, Marc
Access Online
- Series
- Language Note
- English.
- Contents
- Preface to the Second Edition -- Preface to the First Edition -- 1. Measure theory and probability -- 2. Independence and conditioning -- 3. Gaussian variables -- 4. Distributional computations -- 5. Convergence of random variables -- 6. Random processes -- Where is the notion N discussed? -- Final suggestions: how to go further?, Cover -- Exercises in Probability -- Series -- Title -- Copyright -- Dedication -- Contents -- Preface to the second edition -- Preface to the first edition -- Some frequently used notations -- Chapter 1: Measure theory and probability -- 1.1 Some traps concerning the union of σ-fields -- 1.2 Sets which do not belong in a strong sense, to a σ-field -- 1.3 Some criteria for uniform integrability -- 1.4 When does weak convergence imply the convergence of expectations-- 1.5 Conditional expectation and the Monotone Class Theorem -- 1.6 Lp-convergence of conditional expectations -- 1.7 Measure preserving transformations -- 1.8 Ergodic transformations -- 1.9 Invariant σ-fields -- 1.10 Extremal solutions of (general) moments problems -- 1.11 The log normal distribution is moments indeterminate -- 1.12 Conditional expectations and equality in law -- 1.13 Simplifiable random variables -- 1.14 Mellin transform and simplification -- 1.15 There exists no fractional covering of the real line -- Solutions for Chapter 1 -- Solution to Exercise 1.1 -- Solution to Exercise 1.2 -- Solution to Exercise 1.3 -- Solution to Exercise 1.4 -- Solution to Exercise 1.5 -- Solution to Exercise 1.6 -- Solution to Exercise 1.7 -- Solution to Exercise 1.8 -- Solution to Exercise 1.9 -- Solution to Exercise 1.10 -- Solution to Exercise 1.11 -- Solution to Exercise 1.12 -- Solution to Exercise 1.13 -- Solution to Exercise 1.14 -- Solution to Exercise 1.15 -- Chapter 2: Independence and conditioning -- 2.1 Independence does not imply measurability with respect to an independent 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 On exchangeable σ-fields -- 2.8 Too much independence implies constancy -- 2.9 A double paradoxical inequality -- 2.10 Euler's formula for primes and probability -- 2.11 The probability, for integers, of being relatively prime -- 2.12 Completely independent multiplicative sequences of U-valued random variables -- 2.13 Bernoulli random walks considered at some stopping time -- 2.14 cosh, sinh, the Fourier transform and conditional independence -- 2.15 cosh, sinh, and the Laplace transform -- 2.16 Conditioning and changes of probabilities -- 2.17 Radon-Nikodym density and the Acceptance- Rejection Method of von Neumann -- 2.18 Negligible sets and conditioning -- 2.19 Gamma laws and conditioning -- 2.20 Random variables with independent fractional and integer parts -- 2.21 Two characterizations of the simple random walk -- Solutions for Chapter 2 -- Solution to Exercise 2.1 -- Solution to Exercise 2.2 -- Solution to Exercise 2.3 -- Solution to Exercise 2.4 -- Solution to Exercise 2.5 -- Solution to Exercise 2.6 -- Solution to Exercise 2.7 -- Solution to Exercise 2.8 -- Solution to Exercise 2.9 -- 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 -- Solution to Exercise 2.19 -- Solution to Exercise 2.20 -- Solution to Exercise 2.21 -- Chapter 3: Gaussian variables -- 3.1 Constructing Gaussian variables from, but not belonging to, a Gaussian space -- 3.2 A complement to Exercise 3.1 -- 3.3 Gaussian vectors and orthogonal projections -- 3.4 On the negative moments of norms of Gaussian vectors -- 3.5 Quadratic functionals of Gaussian vectors and continued fractions., and 5.11 Convergence in law of stable(μ) variables, as μτ̔̈"»9"·0 -- 5.12 Finite-dimensional convergence in law towards Brownian motion -- 5.13 The empirical process and the Brownian bridge -- 5.14 The functional law of large numbers -- 5.15 The Poisson process and Brownian motion -- 5.16 Brownian bridges converging in law to Brownian motions -- 5.17 An almost sure convergence result for sums of stable random variables -- Chapter 6: Random processes -- 6.1 Jeulin's lemma deals with the absolute convergence of integrals of random processes -- 6.2 Functions of Brownian motion as solutions to SDEs -- the example of (x) = sinh(x) -- 6.3 Bougerol's identity and some Bessel variants -- 6.4 Doléans-Dade exponentials and the Maruyama- Girsanov-Van Schuppen-Wong theorem revisited -- 6.5 The range process of Brownian motion -- 6.6 Symmetric Lévy processes reflected at their minimum and maximum -- E. Csáki's formulae for the ratio of Brownian extremes -- 6.7 Infinite divisibility with respect to time -- 6.8 A toy example for Westwater's renormalization -- 6.9 Some asymptotic laws of planar Brownian motion -- 6.10 Windings of the three-dimensional Brownian motion around a line -- 6.11 Cyclic exchangeability property and uniform law related to the Brownian bridge -- 6.12 Local time and hitting time distributions for the Brownian bridge -- 6.13 Partial absolute continuity of the Brownian bridge distribution with respect to the Brownian distribution -- 6.14 A Brownian interpretation of the duplication formula for the gamma function -- 6.15 Some deterministic time-changes of Brownian motion -- 6.16 A new path construction of Brownian and Bessel bridges -- 6.17 Random scaling of the Brownian bridge -- 6.18 Time-inversion and quadratic functionals of Brownian motion -- Lévy's stochastic area formula -- 6.19 Quadratic variation and local time of semimartingales.
- Summary
- "Derived from extensive teaching experience in Paris, this second edition now includes over 100 exercises in probability. New exercises have been added to reflect important areas of current research in probability theory, including infinite divisibility of stochastic processes, past-future martingales and fluctuation theory. For each exercise the authors provide detailed solutions as well as references for preliminary and further reading. There are also many insightful notes to motivate the student and set the exercises in context"--
- Subject(s)
- Genre(s)
- ISBN
- 9781139526562 (electronic bk.)
1139526561 (electronic bk.)
9781139135351
113913535X
9781283528511
1283528517
9781139531238
1139531239
9781107606555 (Paper)
1107606551 (Paper)
1107232309
9781107232303
1139528955
9781139528955
1139532421
9781139532426
1139527762
9781139527767
9786613840967
6613840963 - Digital File Characteristics
- data file
- Bibliography Note
- Includes bibliographical references (pages 270-277) and index.
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