Quantitative CLTs on a Gaussian space: a survey of recent developments. (English) Zbl 1327.60062
Summary: I will provide a short survey of recent findings concerning normal approximations on a Gaussian space. The results discussed in this work involve Stein’s method, Poincaré-type inequalities, as well as the use of techniques from information theory. The guiding example involves
“exploding Brownian functionals”, that are used as a tool for enhancing the reader’s intuition.
MSC:
| 60F05 | Central limit and other weak theorems |
| 60G15 | Gaussian processes |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
Keywords:
central limit theorems; Gaussian space; normal approximations; Stein’s method; Carbery-Wright inequalities; Poincaré inequalities; Malliavin calculus; relative entropyReferences:
| [1] | C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer (2000). Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris. With a preface by Dominique Bakry and Michel Ledoux. |
| [2] | E. Azmoodeh, S. Campese and G. Poly (2013). Fourth moment theorems for Markov diffusion generators. Preprint. · Zbl 1292.60078 |
| [3] | E. Azmoodeh, D. Malicet and G. Poly (2013). Generalization of the Nualart-Peccati criterion. · Zbl 1342.60026 |
| [4] | A. R. Barron (1986). Entropy and the central limit theorem. Ann. Probab., 14(1), 336–342. · Zbl 0599.60024 |
| [5] | S. Bourguin and G. Peccati (2012). Portmanteau inequalities on the Poisson space: mixed limits and multidimensional clustering. Preprint. · Zbl 1316.60089 |
| [6] | A. Carbery and J. Wright (2001). Distributional and Lqnorm inequalities for polynomials over convex bodies inRn. Math. Res. Lett., 8(3), 233–248. · Zbl 0989.26010 |
| [7] | S. Chatterjee (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields, 143(12), 1–40. · Zbl 1152.60024 |
| [8] | S. Chatterjee and E. Meckes (2008). Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat., 4, 257–283. · Zbl 1162.60310 |
| [9] | L. H. Y. Chen, L. Goldstein, and Q.-M. Shao (2011). Normal approximation by Stein’s method. Probability and its Applications (New York). Springer, Heidelberg. |
| [10] | H. Chernoff (1981). A note on an inequality involving the normal distribution. Ann. Probab. 9(3), 533–535. · Zbl 0457.60014 |
| [11] | L. Coutin and L. Decreusefond (2012). Stein’s method for Brownian approximations. Preprint. · Zbl 0956.60058 |
| [12] | I. Csiszár. Informationstheoretische Konvergenzbegriffe im Raum der Wahrscheinlichkeitsverteilungen (1962). Magyar Tud. Akad. Mat. Kutató Int. Közl., 7, 137–158. |
| [13] | A. Deya, S. Noreddine, and I. Nourdin (2012). Fourth moment theorem and q-Brownian chaos. Comm. Math. Phys., to appear. · Zbl 1278.46054 |
| [14] | C. Döbler (2012). Stein’s method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Pólya urn model. Preprint. |
| [15] | R. M. Dudley (1989). Real analysis and probability. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA. |
| [16] | L. Goldstein and G. Reinert (2012). Stein’s method for the Beta distribution and the Pólya-Eggenberger Urn. To appear in: Advances in Applied Probability · Zbl 1304.60033 |
| [17] | N. Gozlan and C. Léonard (2010). Transport inequalities - A survey. Markov Processes and Related Fields 16, 635-736. · Zbl 1229.26029 |
| [18] | C. Houdré and V. Pérez-Abreu (1995). Covariance identities and inequalities for func- tionals on Wiener and Poisson spaces. Ann. Probab. 23, 400–419. · Zbl 0831.60029 |
| [19] | Y. Hu, F. Lu and D. Nualart (2013). Convergence of densities of some functionals of Gaussian processes. Preprint. · Zbl 1290.60045 |
| [20] | T. Jeulin and M. Yor (1979). Inégalité de Hardy, semi-martingales et faux-amis. In: Séminaire de Probabilités XIII, LNM 721, Springer-Verlag, Berlin Heidelberg New York, 332–359. |
| [21] | O. Johnson (2004). Information theory and the central limit theorem. Imperial College Press, London. · Zbl 1061.60019 |
| [22] | T. Kemp, I. Nourdin, G. Peccati, and R. Speicher (2012). Wigner chaos and the fourth moment. Ann. Probab., 40(4), 1577– 1635. · Zbl 1277.46033 |
| [23] | S. Kullback (1967). A lower bound for discrimination information in terms of variation. IEEE Trans. Info. Theory, 4. |
| [24] | M. Ledoux (2012). Chaos of a Markov operator and the fourth moment condition. Ann. Probab., 40(6), 2439–2459. · Zbl 1266.60042 |
| [25] | D. Marinucci and G. Peccati (2011). Random fields on the sphere : Representation, limit theorems and cosmological applica- tions, volume 389 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge. · Zbl 1260.60004 |
| [26] | I. Nourdin (2013). Lectures on Gaussian approximations with Malliavin calculus. To appear in: Séminaire de probabilités. · Zbl 1291.60048 |
| [27] | I. Nourdin, D. Nualart, and G. Poly (2012). Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electron. J. Probab., 18(22), 1–19. · Zbl 1285.60053 |
| [28] | I. Nourdin and G. Peccati (2010). Stein’s method and exact Berry-Esseen asymptotics for functionals of Gaussian fields. Ann. Probab. 37(6), 2231–2261. · Zbl 1196.60034 |
| [29] | I. Nourdin and G. Peccati (2010). Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs. ALEA Lat. Am. J. Probab. Math. Stat., 7, 341–375. · Zbl 1276.60026 |
| [30] | I. Nourdin and G. Peccati (2009). Noncentral convergence of multiple integrals. Ann. Probab., 37, 1412–1426. · Zbl 1171.60323 |
| [31] | I. Nourdin and G. Peccati (2012). Normal approximations with Malliavin calculus : from Stein’s method to universality. Cambridge Tracts in Mathematics. Cambridge University Press. · Zbl 1266.60001 |
| [32] | I. Nourdin and G. Peccati (2013). The optimal fourth moment theorem. Preprint. · Zbl 1317.60021 |
| [33] | I. Nourdin, G. Peccati and G. Reinert (2009). Second order Poincaré inequalities and CLTs on Wiener space. Journal of Functional Analysis 257, 593-609. · Zbl 1186.60047 |
| [34] | I. Nourdin, G. Peccati, and G. Reinert (2010). Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab., 38(5), 1947–1985. · Zbl 1246.60039 |
| [35] | I. Nourdin, G. Peccati, and A. Réveillac (2010). Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat., 46(1), 45–58. · Zbl 1196.60035 |
| [36] | I. Nourdin, G. Peccati and Y. Swan (2013). Entropy and the fourth moment phenomenon. Preprint. · Zbl 1292.94010 |
| [37] | I. Nourdin and G. Poly (2013). Convergence in total variation on Wiener chaos. Stochastic Process. Appl., 123, 651–674. · Zbl 1259.60029 |
| [38] | I. Nourdin and J. Rosiński (2013). Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws. Ann. Probab., to appear. · Zbl 1301.60026 |
| [39] | D. Nualart (2006). The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second edition. · Zbl 1099.60003 |
| [40] | D. Nualart and S. Ortiz-Latorre (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Sto- chastic Process. Appl., 118(4), 614–628. · Zbl 1142.60015 |
| [41] | D. Nualart and G. Peccati (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab., 33(1), 177–193. · Zbl 1097.60007 |
| [42] | G. Peccati (2009). Stein’s method, Malliavin calculus and infinite-dimensional Gaussian analysis. Unpublished lecture notes. |
| [43] | G. Peccati, G.; J.L. Solé, M.S. Taqqu and F. Utzet (2010). Stein’s method and normal approximation of Poisson functionals, Ann. Probab. 38, 443–478. · Zbl 1195.60037 |
| [44] | G. Peccati, G. and M.S. Taqqu (2011). Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi University Press and Springer, Milan. · Zbl 1231.60003 |
| [45] | G. Peccati and C.A. Tudor (2005). Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, LNM 1857, 247-262. · Zbl 1063.60027 |
| [46] | G. Peccati and M. Yor (2004). Hardy’s inequality in L2([0, 1]) and principal values of Brownian local times. Asymptotic Methods in Stochastics, AMS, Fields Institute Communications Series, 49–74. · Zbl 1074.60029 |
| [47] | G. Peccati and M. Yor (2004). Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge. Asymptotic Methods in Stochastics, AMS, Fields Institute Communication Series, 75–87. · Zbl 1071.60017 |
| [48] | G. Peccati and C. Zheng (2010). Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab., 15, 1487–1527. · Zbl 1228.60031 |
| [49] | M. S. Pinsker (1964). Information and information stability of random variables and processes. Translated and edited by Amiel Feinstein. Holden-Day Inc., San Francisco, Calif.. · Zbl 0125.09202 |
| [50] | G. Reinert and A. Roellin (2009). Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab., 37(6), 2150–2173. · Zbl 1200.62010 |
| [51] | M. Reitzner and M. Schulte (2012). Central limit theorems for U-statistics of Poisson point processes. Ann. Probab., to appear. · Zbl 1293.60061 |
| [52] | D. Revuz and M. Yor (1999). Continuous martingales and Brownian motion. Springer-Verlag, Berlin Heidelberg New York. · Zbl 0917.60006 |
| [53] | M. Talagrand (1986). Transportation cost for Gaussian and other product measures. Geom. Funct. Anal., 6(3), 587–600, 1996. · Zbl 0859.46030 |
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