Normal Approximations with Malliavin Calculus

Erscheinungsjahr 2015
ISBN: 978-1-107-01777-1
Verlag: Cambridge University Press

Buch, Englisch, Band 192, 254 Seiten, Format (B × H): 157 mm x 235 mm, Gewicht: 568 g

Reihe: Cambridge Tracts in Mathematics

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.

Autoren/Hrsg.

Nourdin, Ivan

Peccati, Giovanni

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung

Inhaltsverzeichnis

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.

Über Autor(innen)

Nourdin, Ivan
Ivan Nourdin is Full Professor at Nancy University 1, France.

Peccati, Giovanni
Giovanni Peccati is Full Professor in Stochastic Analysis and Finance at the University of Luxembourg.