Stein's method is a collection of probabilistic techniques that allow one to assess the distance between two probability distributions by means of differential operators. In 2007, the authors discovered that one can combine Stein's method with the powerful Malliavin calculus of variations, in order to deduce quantitative central limit theorems involving functionals of general Gaussian fields. This book provides an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space. Many recent developments and applications are studied in detail, for instance: fourth moment theorems on the Wiener chaos, density estimates, Breuer–Major theorems for fractional processes, recursive cumulant computations, optimal rates and universality results for homogeneous sums. Largely self-contained, the book is perfect for self-study. It will appeal to researchers and graduate students in probability and statistics, especially those who wish to understand the connections between Stein's method and Malliavin calculus.
- Contains an introduction for readers who are not familiar with Malliavin calculus and/or Stein's method
- Provides the first unified view of two separate fields of research
- Includes detailed proofs
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Awards
- Winner of the 2015 Outstanding Scientific Publication Prize, National Foundation for Science of Luxembourg
Reviews & endorsements
'This monograph is a nice and excellent introduction to Malliavin calculus and its application to deducing quantitative central limit theorems in combination with Stein's method for normal approximation. It provides a self-contained and appealing presentation of the recent work developed by the authors, and it is well tailored for graduate students and researchers.' David Nualart, Mathematical Reviews
'The book contains many examples and exercises which help the reader understand and assimilate the material. Also bibliographical comments at the end of each chapter provide useful references for further reading.' Bulletin of the American Mathematical Society
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Product details
- Date Published: May 2012
- format: Hardback
- isbn: 9781107017771
- length: 254 pages
- dimensions: 229 x 152 x 18 mm
- weight: 0.49kg
- contains: 70 exercises
- availability: Available
Table of Contents
Preface
Introduction
1. Malliavin operators in the one-dimensional case
2. Malliavin operators and isonormal Gaussian processes
3. Stein's method for one-dimensional normal approximations
4. Multidimensional Stein's method
5. Stein meets Malliavin: univariate normal approximations
6. Multivariate normal approximations
7. Exploring the Breuer–Major Theorem
8. Computation of cumulants
9. Exact asymptotics and optimal rates
10. Density estimates
11. Homogeneous sums and universality
Appendix 1. Gaussian elements, cumulants and Edgeworth expansions
Appendix 2. Hilbert space notation
Appendix 3. Distances between probability measures
Appendix 4. Fractional Brownian motion
Appendix 5. Some results from functional analysis
References
Index.
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Normal Approximations with Malliavin Calculus

Ivan Nourdin, Giovanni Peccati
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Authors
Ivan Nourdin, Université de Nancy I, France
Ivan Nourdin is Full Professor at Nancy University 1, France.
Giovanni Peccati, Université du Luxembourg
Giovanni Peccati is Full Professor in Stochastic Analysis and Finance at the University of Luxembourg.
Awards
- Winner of the 2015 Outstanding Scientific Publication Prize, National Foundation for Science of Luxembourg