Normal approximations with Malliavin calculus: from Stein's method to universality
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,...
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2012
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| Series: | Cambridge tracts in mathematics
192 |
| Subjects: | |
| Online Access: | BSB01 FHN01 UBR01 Volltext |
| Summary: | 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 |
| Physical Description: | 1 Online-Ressource (xiv, 239 Seiten) |
| ISBN: | 9781139084659 |
| DOI: | 10.1017/CBO9781139084659 |
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| contents | Malliavin operators in the one-dimensional case -- Malliavin operators and isonormal Gaussian processes -- Stein's method for one-dimensional normal approximations -- Multidimensional Stein's method -- Stein meets Malliavin : univariate normal approximations -- Multivariate normal approximations -- Exploring the Breuer-Major theorem -- Computation of cumulants -- Exact asymptotics and optimal rates -- Density estimates -- Homogeneous sums and universality -- Gaussian elements, cumulants and Edgeworth expansions -- Hilbert space notation -- Distances between probability measures -- Fractional Brownian motion -- Some results from functional analysis |
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/3 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV043941784 |
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| indexdate | 2024-07-10T07:39:16Z |
| institution | BVB |
| isbn | 9781139084659 |
| language | English |
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| physical | 1 Online-Ressource (xiv, 239 Seiten) |
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| spelling | Nourdin, Ivan 1978- Verfasser (DE-588)1023866684 aut Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin, Giovanni Peccati Cambridge Cambridge University Press 2012 1 Online-Ressource (xiv, 239 Seiten) txt rdacontent c rdamedia cr rdacarrier Cambridge tracts in mathematics 192 Malliavin operators in the one-dimensional case -- Malliavin operators and isonormal Gaussian processes -- Stein's method for one-dimensional normal approximations -- Multidimensional Stein's method -- Stein meets Malliavin : univariate normal approximations -- Multivariate normal approximations -- Exploring the Breuer-Major theorem -- Computation of cumulants -- Exact asymptotics and optimal rates -- Density estimates -- Homogeneous sums and universality -- Gaussian elements, cumulants and Edgeworth expansions -- Hilbert space notation -- Distances between probability measures -- Fractional Brownian motion -- Some results from functional analysis 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 Approximation theory Malliavin calculus Approximation (DE-588)4002498-2 gnd rswk-swf Malliavin-Kalkül (DE-588)4242584-0 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd rswk-swf Stein-Schätzung (DE-588)7570767-6 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 s Approximation (DE-588)4002498-2 s Malliavin-Kalkül (DE-588)4242584-0 s Stein-Schätzung (DE-588)7570767-6 s DE-604 Peccati, Giovanni 1975- Sonstige (DE-588)1018525807 oth Erscheint auch als Druck-Ausgabe 978-1-107-01777-1 https://doi.org/10.1017/CBO9781139084659 Verlag URL des Erstveröffentlichers Volltext |
| spellingShingle | Nourdin, Ivan 1978- Normal approximations with Malliavin calculus from Stein's method to universality Malliavin operators in the one-dimensional case -- Malliavin operators and isonormal Gaussian processes -- Stein's method for one-dimensional normal approximations -- Multidimensional Stein's method -- Stein meets Malliavin : univariate normal approximations -- Multivariate normal approximations -- Exploring the Breuer-Major theorem -- Computation of cumulants -- Exact asymptotics and optimal rates -- Density estimates -- Homogeneous sums and universality -- Gaussian elements, cumulants and Edgeworth expansions -- Hilbert space notation -- Distances between probability measures -- Fractional Brownian motion -- Some results from functional analysis Approximation theory Malliavin calculus Approximation (DE-588)4002498-2 gnd Malliavin-Kalkül (DE-588)4242584-0 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd Stein-Schätzung (DE-588)7570767-6 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)4242584-0 (DE-588)4121894-2 (DE-588)7570767-6 |
| title | Normal approximations with Malliavin calculus from Stein's method to universality |
| title_auth | Normal approximations with Malliavin calculus from Stein's method to universality |
| title_exact_search | Normal approximations with Malliavin calculus from Stein's method to universality |
| title_full | Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin, Giovanni Peccati |
| title_fullStr | Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin, Giovanni Peccati |
| title_full_unstemmed | Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin, Giovanni Peccati |
| title_short | Normal approximations with Malliavin calculus |
| title_sort | normal approximations with malliavin calculus from stein s method to universality |
| title_sub | from Stein's method to universality |
| topic | Approximation theory Malliavin calculus Approximation (DE-588)4002498-2 gnd Malliavin-Kalkül (DE-588)4242584-0 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd Stein-Schätzung (DE-588)7570767-6 gnd |
| topic_facet | Approximation theory Malliavin calculus Approximation Malliavin-Kalkül Wahrscheinlichkeitsverteilung Stein-Schätzung |
| url | https://doi.org/10.1017/CBO9781139084659 |
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