Normal Approximations with Malliavin Calculus :: From Stein's Method to Universality.
Shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus.
Gespeichert in:
| 1. Verfasser: | |
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| Weitere Verfasser: | |
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge :
Cambridge University Press,
2012.
|
| Schriftenreihe: | Cambridge Tracts in Mathematics, 192.
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | Shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus. |
| Beschreibung: | 11.6 Exercises. |
| Beschreibung: | 1 online resource (256 pages) |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9781139377355 1139377353 9781139084659 1139084658 9781139380218 1139380214 9781139375924 113937592X |
Internformat
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| 100 | 1 | |a Nourdin, Ivan. | |
| 245 | 1 | 0 | |a Normal Approximations with Malliavin Calculus : |b From Stein's Method to Universality. |
| 260 | |a Cambridge : |b Cambridge University Press, |c 2012. | ||
| 300 | |a 1 online resource (256 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Cambridge Tracts in Mathematics, 192 ; |v v. 192 | |
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Cover; CAMBRIDGE TRACTS IN MATHEMATICS: GENERAL EDITORS; Title; Copyright; Dedication; Contents; Preface; Introduction; 1 Malliavin operators in the one-dimensional case; 1.1 Derivative operators; 1.2 Divergences; 1.3 Ornstein-Uhlenbeck operators; 1.4 First application: Hermite polynomials; 1.5 Second application: variance expansions; 1.6 Third application: second-order Poincaré inequalities; 1.7 Exercises; 1.8 Bibliographic comments; 2 Malliavin operators and isonormal Gaussian processes; 2.1 Isonormal Gaussian processes; 2.2 Wiener chaos; 2.3 The derivative operator. | |
| 505 | 8 | |a 2.4 The Malliavin derivatives in Hilbert spaces2.5 The divergence operator; 2.6 Some Hilbert space valued divergences; 2.7 Multiple integrals; 2.8 The Ornstein-Uhlenbeck semigroup; 2.9 An integration by parts formula; 2.10 Absolute continuity of the laws of multiple integrals; 2.11 Exercises; 2.12 Bibliographic comments; 3 Stein's method for one-dimensional normal approximations; 3.1 Gaussian moments and Stein's lemma; 3.2 Stein's equations; 3.3 Stein's bounds for the total variation distance; 3.4 Stein's bounds for the Kolmogorov distance; 3.5 Stein's bounds for the Wasserstein distance. | |
| 505 | 8 | |a 3.6 A simple example3.7 The Berry-Esseen theorem; 3.8 Exercises; 3.9 Bibliographic comments; 4 Multidimensional Stein's method; 4.1 Multidimensional Stein's lemmas; 4.2 Stein's equations for identity matrices; 4.3 Stein's equations for general positive definite matrices; 4.4 Bounds on the Wasserstein distance; 4.5 Exercises; 4.6 Bibliographic comments; 5 Stein meets Malliavin: univariate normal approximations; 5.1 Bounds for general functionals; 5.2 Normal approximations on Wiener chaos; 5.2.1 Some preliminary considerations; 5.3 Normal approximations in the general case; 5.3.1 Main results. | |
| 505 | 8 | |a 5.4 Exercises5.5 Bibliographic comments; 6 Multivariate normal approximations; 6.1 Bounds for general vectors; 6.2 The case of Wiener chaos; 6.3 CLTs via chaos decompositions; 6.4 Exercises; 6.5 Bibliographic comments; 7 Exploring the Breuer-Major theorem; 7.1 Motivation; 7.2 A general statement; 7.3 Quadratic case; 7.4 The increments of a fractional Brownian motion; 7.5 Exercises; 7.6 Bibliographic comments; 8 Computation of cumulants; 8.1 Decomposing multi-indices; 8.2 General formulae; 8.3 Application to multiple integrals; 8.4 Formulae in dimension one; 8.5 Exercises. | |
| 505 | 8 | |a 8.6 Bibliographic comments9 Exact asymptotics and optimal rates; 9.1 Some technical computations; 9.2 A general result; 9.3 Connections with Edgeworth expansions; 9.4 Double integrals; 9.5 Further examples; 9.6 Exercises; 9.7 Bibliographic comments; 10 Density estimates; 10.1 General results; 10.2 Explicit computations; 10.3 An example; 10.4 Exercises; 10.5 Bibliographic comments; 11 Homogeneous sums and universality; 11.1 The Lindeberg method; 11.2 Homogeneous sums and influence functions; 11.3 The universality result; 11.4 Some technical estimates; 11.5 Proof of Theorem 11.3.1. | |
| 500 | |a 11.6 Exercises. | ||
| 520 | |a Shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus. | ||
| 504 | |a Includes bibliographical references and index. | ||
| 650 | 0 | |a Approximation theory. |0 http://id.loc.gov/authorities/subjects/sh85006190 | |
| 650 | 0 | |a Malliavin calculus. |0 http://id.loc.gov/authorities/subjects/sh91004306 | |
| 650 | 6 | |a Théorie de l'approximation. | |
| 650 | 6 | |a Calcul de Malliavin. | |
| 650 | 7 | |a MATHEMATICS |x Probability & Statistics |x General. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Probability & Statistics |x Stochastic Processes. |2 bisacsh | |
| 650 | 0 | 7 | |a Aproximación, Teoría de |2 embucm |
| 650 | 7 | |a Approximation theory |2 fast | |
| 650 | 7 | |a Malliavin calculus |2 fast | |
| 700 | 1 | |a Peccati, Giovanni, |d 1975- |1 https://id.oclc.org/worldcat/entity/E39PCjHW3xR6VdxggCgh4BRBT3 |0 http://id.loc.gov/authorities/names/n2011037471 | |
| 758 | |i has work: |a Normal approximations with Malliavin calculus (Text) |1 https://id.oclc.org/worldcat/entity/E39PCG3RkMDtRqjXjvR8WFHq6X |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Nourdin, Ivan. |t Normal Approximations with Malliavin Calculus : From Stein's Method to Universality. |d Cambridge : Cambridge University Press, ©2012 |z 9781107017771 |
| 830 | 0 | |a Cambridge Tracts in Mathematics, 192. | |
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Datensatz im Suchindex
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn797919775 |
|---|---|
| _version_ | 1816882199566221312 |
| adam_text | |
| any_adam_object | |
| author | Nourdin, Ivan |
| author2 | Peccati, Giovanni, 1975- |
| author2_role | |
| author2_variant | g p gp |
| author_GND | http://id.loc.gov/authorities/names/n2011037471 |
| author_facet | Nourdin, Ivan Peccati, Giovanni, 1975- |
| author_role | |
| author_sort | Nourdin, Ivan |
| author_variant | i n in |
| building | Verbundindex |
| bvnumber | localFWS |
| callnumber-first | Q - Science |
| callnumber-label | QA221 |
| callnumber-raw | QA221 .N68 2012 |
| callnumber-search | QA221 .N68 2012 |
| callnumber-sort | QA 3221 N68 42012 |
| callnumber-subject | QA - Mathematics |
| collection | ZDB-4-EBA |
| contents | Cover; CAMBRIDGE TRACTS IN MATHEMATICS: GENERAL EDITORS; Title; Copyright; Dedication; Contents; Preface; Introduction; 1 Malliavin operators in the one-dimensional case; 1.1 Derivative operators; 1.2 Divergences; 1.3 Ornstein-Uhlenbeck operators; 1.4 First application: Hermite polynomials; 1.5 Second application: variance expansions; 1.6 Third application: second-order Poincaré inequalities; 1.7 Exercises; 1.8 Bibliographic comments; 2 Malliavin operators and isonormal Gaussian processes; 2.1 Isonormal Gaussian processes; 2.2 Wiener chaos; 2.3 The derivative operator. 2.4 The Malliavin derivatives in Hilbert spaces2.5 The divergence operator; 2.6 Some Hilbert space valued divergences; 2.7 Multiple integrals; 2.8 The Ornstein-Uhlenbeck semigroup; 2.9 An integration by parts formula; 2.10 Absolute continuity of the laws of multiple integrals; 2.11 Exercises; 2.12 Bibliographic comments; 3 Stein's method for one-dimensional normal approximations; 3.1 Gaussian moments and Stein's lemma; 3.2 Stein's equations; 3.3 Stein's bounds for the total variation distance; 3.4 Stein's bounds for the Kolmogorov distance; 3.5 Stein's bounds for the Wasserstein distance. 3.6 A simple example3.7 The Berry-Esseen theorem; 3.8 Exercises; 3.9 Bibliographic comments; 4 Multidimensional Stein's method; 4.1 Multidimensional Stein's lemmas; 4.2 Stein's equations for identity matrices; 4.3 Stein's equations for general positive definite matrices; 4.4 Bounds on the Wasserstein distance; 4.5 Exercises; 4.6 Bibliographic comments; 5 Stein meets Malliavin: univariate normal approximations; 5.1 Bounds for general functionals; 5.2 Normal approximations on Wiener chaos; 5.2.1 Some preliminary considerations; 5.3 Normal approximations in the general case; 5.3.1 Main results. 5.4 Exercises5.5 Bibliographic comments; 6 Multivariate normal approximations; 6.1 Bounds for general vectors; 6.2 The case of Wiener chaos; 6.3 CLTs via chaos decompositions; 6.4 Exercises; 6.5 Bibliographic comments; 7 Exploring the Breuer-Major theorem; 7.1 Motivation; 7.2 A general statement; 7.3 Quadratic case; 7.4 The increments of a fractional Brownian motion; 7.5 Exercises; 7.6 Bibliographic comments; 8 Computation of cumulants; 8.1 Decomposing multi-indices; 8.2 General formulae; 8.3 Application to multiple integrals; 8.4 Formulae in dimension one; 8.5 Exercises. 8.6 Bibliographic comments9 Exact asymptotics and optimal rates; 9.1 Some technical computations; 9.2 A general result; 9.3 Connections with Edgeworth expansions; 9.4 Double integrals; 9.5 Further examples; 9.6 Exercises; 9.7 Bibliographic comments; 10 Density estimates; 10.1 General results; 10.2 Explicit computations; 10.3 An example; 10.4 Exercises; 10.5 Bibliographic comments; 11 Homogeneous sums and universality; 11.1 The Lindeberg method; 11.2 Homogeneous sums and influence functions; 11.3 The universality result; 11.4 Some technical estimates; 11.5 Proof of Theorem 11.3.1. |
| ctrlnum | (OCoLC)797919775 |
| dewey-full | 519.2/3 519.23 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/3 519.23 |
| dewey-search | 519.2/3 519.23 |
| dewey-sort | 3519.2 13 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn797919775 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:24:49Z |
| institution | BVB |
| isbn | 9781139377355 1139377353 9781139084659 1139084658 9781139380218 1139380214 9781139375924 113937592X |
| language | English |
| oclc_num | 797919775 |
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| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (256 pages) |
| psigel | ZDB-4-EBA |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | Cambridge Tracts in Mathematics, 192. |
| series2 | Cambridge Tracts in Mathematics, 192 ; |
| spelling | Nourdin, Ivan. Normal Approximations with Malliavin Calculus : From Stein's Method to Universality. Cambridge : Cambridge University Press, 2012. 1 online resource (256 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Cambridge Tracts in Mathematics, 192 ; v. 192 Print version record. Cover; CAMBRIDGE TRACTS IN MATHEMATICS: GENERAL EDITORS; Title; Copyright; Dedication; Contents; Preface; Introduction; 1 Malliavin operators in the one-dimensional case; 1.1 Derivative operators; 1.2 Divergences; 1.3 Ornstein-Uhlenbeck operators; 1.4 First application: Hermite polynomials; 1.5 Second application: variance expansions; 1.6 Third application: second-order Poincaré inequalities; 1.7 Exercises; 1.8 Bibliographic comments; 2 Malliavin operators and isonormal Gaussian processes; 2.1 Isonormal Gaussian processes; 2.2 Wiener chaos; 2.3 The derivative operator. 2.4 The Malliavin derivatives in Hilbert spaces2.5 The divergence operator; 2.6 Some Hilbert space valued divergences; 2.7 Multiple integrals; 2.8 The Ornstein-Uhlenbeck semigroup; 2.9 An integration by parts formula; 2.10 Absolute continuity of the laws of multiple integrals; 2.11 Exercises; 2.12 Bibliographic comments; 3 Stein's method for one-dimensional normal approximations; 3.1 Gaussian moments and Stein's lemma; 3.2 Stein's equations; 3.3 Stein's bounds for the total variation distance; 3.4 Stein's bounds for the Kolmogorov distance; 3.5 Stein's bounds for the Wasserstein distance. 3.6 A simple example3.7 The Berry-Esseen theorem; 3.8 Exercises; 3.9 Bibliographic comments; 4 Multidimensional Stein's method; 4.1 Multidimensional Stein's lemmas; 4.2 Stein's equations for identity matrices; 4.3 Stein's equations for general positive definite matrices; 4.4 Bounds on the Wasserstein distance; 4.5 Exercises; 4.6 Bibliographic comments; 5 Stein meets Malliavin: univariate normal approximations; 5.1 Bounds for general functionals; 5.2 Normal approximations on Wiener chaos; 5.2.1 Some preliminary considerations; 5.3 Normal approximations in the general case; 5.3.1 Main results. 5.4 Exercises5.5 Bibliographic comments; 6 Multivariate normal approximations; 6.1 Bounds for general vectors; 6.2 The case of Wiener chaos; 6.3 CLTs via chaos decompositions; 6.4 Exercises; 6.5 Bibliographic comments; 7 Exploring the Breuer-Major theorem; 7.1 Motivation; 7.2 A general statement; 7.3 Quadratic case; 7.4 The increments of a fractional Brownian motion; 7.5 Exercises; 7.6 Bibliographic comments; 8 Computation of cumulants; 8.1 Decomposing multi-indices; 8.2 General formulae; 8.3 Application to multiple integrals; 8.4 Formulae in dimension one; 8.5 Exercises. 8.6 Bibliographic comments9 Exact asymptotics and optimal rates; 9.1 Some technical computations; 9.2 A general result; 9.3 Connections with Edgeworth expansions; 9.4 Double integrals; 9.5 Further examples; 9.6 Exercises; 9.7 Bibliographic comments; 10 Density estimates; 10.1 General results; 10.2 Explicit computations; 10.3 An example; 10.4 Exercises; 10.5 Bibliographic comments; 11 Homogeneous sums and universality; 11.1 The Lindeberg method; 11.2 Homogeneous sums and influence functions; 11.3 The universality result; 11.4 Some technical estimates; 11.5 Proof of Theorem 11.3.1. 11.6 Exercises. Shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus. Includes bibliographical references and index. Approximation theory. http://id.loc.gov/authorities/subjects/sh85006190 Malliavin calculus. http://id.loc.gov/authorities/subjects/sh91004306 Théorie de l'approximation. Calcul de Malliavin. MATHEMATICS Probability & Statistics General. bisacsh MATHEMATICS Probability & Statistics Stochastic Processes. bisacsh Aproximación, Teoría de embucm Approximation theory fast Malliavin calculus fast Peccati, Giovanni, 1975- https://id.oclc.org/worldcat/entity/E39PCjHW3xR6VdxggCgh4BRBT3 http://id.loc.gov/authorities/names/n2011037471 has work: Normal approximations with Malliavin calculus (Text) https://id.oclc.org/worldcat/entity/E39PCG3RkMDtRqjXjvR8WFHq6X https://id.oclc.org/worldcat/ontology/hasWork Print version: Nourdin, Ivan. Normal Approximations with Malliavin Calculus : From Stein's Method to Universality. Cambridge : Cambridge University Press, ©2012 9781107017771 Cambridge Tracts in Mathematics, 192. FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=443707 Volltext |
| spellingShingle | Nourdin, Ivan Normal Approximations with Malliavin Calculus : From Stein's Method to Universality. Cambridge Tracts in Mathematics, 192. Cover; CAMBRIDGE TRACTS IN MATHEMATICS: GENERAL EDITORS; Title; Copyright; Dedication; Contents; Preface; Introduction; 1 Malliavin operators in the one-dimensional case; 1.1 Derivative operators; 1.2 Divergences; 1.3 Ornstein-Uhlenbeck operators; 1.4 First application: Hermite polynomials; 1.5 Second application: variance expansions; 1.6 Third application: second-order Poincaré inequalities; 1.7 Exercises; 1.8 Bibliographic comments; 2 Malliavin operators and isonormal Gaussian processes; 2.1 Isonormal Gaussian processes; 2.2 Wiener chaos; 2.3 The derivative operator. 2.4 The Malliavin derivatives in Hilbert spaces2.5 The divergence operator; 2.6 Some Hilbert space valued divergences; 2.7 Multiple integrals; 2.8 The Ornstein-Uhlenbeck semigroup; 2.9 An integration by parts formula; 2.10 Absolute continuity of the laws of multiple integrals; 2.11 Exercises; 2.12 Bibliographic comments; 3 Stein's method for one-dimensional normal approximations; 3.1 Gaussian moments and Stein's lemma; 3.2 Stein's equations; 3.3 Stein's bounds for the total variation distance; 3.4 Stein's bounds for the Kolmogorov distance; 3.5 Stein's bounds for the Wasserstein distance. 3.6 A simple example3.7 The Berry-Esseen theorem; 3.8 Exercises; 3.9 Bibliographic comments; 4 Multidimensional Stein's method; 4.1 Multidimensional Stein's lemmas; 4.2 Stein's equations for identity matrices; 4.3 Stein's equations for general positive definite matrices; 4.4 Bounds on the Wasserstein distance; 4.5 Exercises; 4.6 Bibliographic comments; 5 Stein meets Malliavin: univariate normal approximations; 5.1 Bounds for general functionals; 5.2 Normal approximations on Wiener chaos; 5.2.1 Some preliminary considerations; 5.3 Normal approximations in the general case; 5.3.1 Main results. 5.4 Exercises5.5 Bibliographic comments; 6 Multivariate normal approximations; 6.1 Bounds for general vectors; 6.2 The case of Wiener chaos; 6.3 CLTs via chaos decompositions; 6.4 Exercises; 6.5 Bibliographic comments; 7 Exploring the Breuer-Major theorem; 7.1 Motivation; 7.2 A general statement; 7.3 Quadratic case; 7.4 The increments of a fractional Brownian motion; 7.5 Exercises; 7.6 Bibliographic comments; 8 Computation of cumulants; 8.1 Decomposing multi-indices; 8.2 General formulae; 8.3 Application to multiple integrals; 8.4 Formulae in dimension one; 8.5 Exercises. 8.6 Bibliographic comments9 Exact asymptotics and optimal rates; 9.1 Some technical computations; 9.2 A general result; 9.3 Connections with Edgeworth expansions; 9.4 Double integrals; 9.5 Further examples; 9.6 Exercises; 9.7 Bibliographic comments; 10 Density estimates; 10.1 General results; 10.2 Explicit computations; 10.3 An example; 10.4 Exercises; 10.5 Bibliographic comments; 11 Homogeneous sums and universality; 11.1 The Lindeberg method; 11.2 Homogeneous sums and influence functions; 11.3 The universality result; 11.4 Some technical estimates; 11.5 Proof of Theorem 11.3.1. Approximation theory. http://id.loc.gov/authorities/subjects/sh85006190 Malliavin calculus. http://id.loc.gov/authorities/subjects/sh91004306 Théorie de l'approximation. Calcul de Malliavin. MATHEMATICS Probability & Statistics General. bisacsh MATHEMATICS Probability & Statistics Stochastic Processes. bisacsh Aproximación, Teoría de embucm Approximation theory fast Malliavin calculus fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85006190 http://id.loc.gov/authorities/subjects/sh91004306 |
| 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. |
| title_fullStr | Normal Approximations with Malliavin Calculus : From Stein's Method to Universality. |
| title_full_unstemmed | Normal Approximations with Malliavin Calculus : From Stein's Method to Universality. |
| 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. http://id.loc.gov/authorities/subjects/sh85006190 Malliavin calculus. http://id.loc.gov/authorities/subjects/sh91004306 Théorie de l'approximation. Calcul de Malliavin. MATHEMATICS Probability & Statistics General. bisacsh MATHEMATICS Probability & Statistics Stochastic Processes. bisacsh Aproximación, Teoría de embucm Approximation theory fast Malliavin calculus fast |
| topic_facet | Approximation theory. Malliavin calculus. Théorie de l'approximation. Calcul de Malliavin. MATHEMATICS Probability & Statistics General. MATHEMATICS Probability & Statistics Stochastic Processes. Aproximación, Teoría de Approximation theory Malliavin calculus |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=443707 |
| work_keys_str_mv | AT nourdinivan normalapproximationswithmalliavincalculusfromsteinsmethodtouniversality AT peccatigiovanni normalapproximationswithmalliavincalculusfromsteinsmethodtouniversality |