Malliavin–Stein method: a survey of some recent developments

Azmoodeh, Ehsan; Peccati, Giovanni; Yang, Xiaochuan

doi:10.15559/21-VMSTA184

Modern Stochastics: Theory and Applications

Malliavin–Stein method: a survey of some recent developments

Volume 8, Issue 2 (2021), pp. 141–177

Ehsan Azmoodeh Giovanni Peccati Xiaochuan Yang

https://doi.org/10.15559/21-VMSTA184

Pub. online: 22 June 2021 Type: Research Article

Open Access

Received
12 February 2021

Revised
28 May 2021

Accepted
10 June 2021

Published
22 June 2021

Abstract

Initiated around the year 2007, the Malliavin–Stein approach to probabilistic approximations combines Stein’s method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin–Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin–Stein method, with specific emphasis on extensions and generalizations in the framework of Markov semigroups and of random point measures.

References

[1]

A webpage about Stein’s method and Malliavin calculus (by I. Nourdin). https://sites.google.com/site/malliavinstein/home

[2]

Arras, B., Azmoodeh, E., Poly, G., Swan, Y.: A bound on the 2-Wasserstein distance between linear combinations of independent random variables. Stoch. Process. Appl. 129(7), 2341–2375 (2019). MR3958435. https://doi.org/10.1016/j.spa.2018.07.009

[3]

Arras, B., Azmoodeh, E., Poly, G., Swan, Y.: Stein characterizations for linear combinations of gamma random variables. Braz. J. Probab. Stat. 34(2), 394–413 (2020). MR4093265. https://doi.org/10.1214/18-BJPS420

[4]

Arras, B., Mijoule, G., Poly, G., Swan, Y.: A new approach to the Stein-Tikhomirov method: with applications to the second Wiener chaos and Dickman convergence (2016). arXiv:1605.06819

[5]

Azmoodeh, E., Campese, S., Poly, G.: Fourth Moment Theorems for Markov diffusion generators. J. Funct. Anal. 266(4), 2341–2359 (2014). MR3150163. https://doi.org/10.1016/j.jfa.2013.10.014

[6]

Azmoodeh, E., Eichelsbacher, P., Knichel, L.: Optimal gamma approximation on Wiener space. ALEA Lat. Am. J. Probab. Math. Stat. 17(1), 101–132 (2020). MR4057185. https://doi.org/10.30757/alea.v17-05

[7]

Azmoodeh, E., Gasbarra, D.: New moments criteria for convergence towards normal product/tetilla laws (2017). arXiv:1708.07681

[8]

Azmoodeh, E., Malicet, D., Mijoule, G., Poly, G.: Generalization of the Nualart-Peccati criterion. Ann. Probab. 44(2), 924–954 (2016). MR3474463. https://doi.org/10.1214/14-AOP992

[9]

Azmoodeh, E., Peccati, G., Poly, G.: Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach. In: In memoriam Marc Yor—Séminaire de Probabilités XLVII. Lecture Notes in Math., vol. 2137, pp. 339–367. Springer, Cham (2015). MR3444306. https://doi.org/10.1007/978-3-319-18585-9_16

[10]

Bakry, D.: L‘hypercontractivité et son utilisation en théorie des semigroupes. In: Lectures on probability theory, Saint-Flour, 1992. Lecture Notes in Math., vol. 1581, pp. 1–114. Springer, Berlin (1994). MR1307413. https://doi.org/10.1007/BFb0073872

[11]

Barbour, A.D.: Stein’s method for diffusion approximations. Probab. Theory Relat. Fields 84(3), 297–322 (1990). MR1035659. https://doi.org/10.1007/BF01197887

[12]

Barbour, A.D., Janson, S.: A functional combinatorial central limit theorem. Electron. J. Probab. 14, 2352–2370 (2009). MR2556014. https://doi.org/10.1214/EJP.v14-709

[13]

Baryshnikov, Yu., Yukich, J.E.: Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15(1A), 213–253 (2005). MR2115042. https://doi.org/10.1214/105051604000000594

[14]

Biermé, H., Bonami, A., Nourdin, I., Peccati, G.: Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants. ALEA Lat. Am. J. Probab. Math. Stat. 9(2), 473–500 (2012). MR3069374

[15]

Bonis, Th.: Stein’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem. Probab. Theory Relat. Fields 178, 827–860 (2020). MR4168389. https://doi.org/10.1007/s00440-020-00989-4

[16]

Bourguin, S., Campese, S., Leonenko, N., Taqqu, M.S.: Four moments theorems on Markov chaos. Ann. Probab. 47(3), 1417–1446 (2019). MR3945750. https://doi.org/10.1214/18-AOP1287

[17]

Bourguin, S., Campese, S.: Approximation of Hilbert-Valued Gaussians on Dirichlet structures. Electron. J. Probab. 25, 150 (2020), 30 pp. MR4193891. https://doi.org/10.1214/20-ejp551

[18]

Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Springer (2014). MR3155209. https://doi.org/10.1007/978-3-319-00227-9

[19]

Campese, S., Nourdin, I., Nualart, D.: Continuous Breuer-Major theorem: tightness and non-stationarity. Ann. Probab. 48(1), 147–177 (2020). MR4079433. https://doi.org/10.1214/19-AOP1357

[20]

Can, V.H., Trinh, K.D.: Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals arXiv:2004.06313 (2020). MR4049088. https://doi.org/10.1214/19-ecp279

[21]

Chatterjee, S.: Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Relat. Fields 143, 1–40 (2009). MR2449121. https://doi.org/10.1007/s00440-007-0118-6

[22]

Chatterjee, S., Sen, S.: Minimal spanning trees and Stein’s method. Ann. Appl. Probab. 27(3), 1588–1645 (2017). MR3678480. https://doi.org/10.1214/16-AAP1239

[23]

Chen, L.H.Y., Goldstein, L., Shao, Q.M.: Normal approximation by Stein’s method. Springer (2010). MR2732624. https://doi.org/10.1007/978-3-642-15007-4

[24]

Chen, L.H., Poly, G.: Stein’s method, Malliavin calculus, Dirichlet forms and the fourth moment theorem. In: Festschrift of Masatoshi Fukushima. Interdisciplinary Mathematical Sciences, vol. 17, pp. 107–130 (2015). MR3379337. https://doi.org/10.1142/9789814596534_0006

[25]

Courtade, T.A., Fathi, M., Pananjady, A.: Existence of Stein kernels under a spectral gap, and discrepancy bounds. Ann. Inst. Henri Poincaré Probab. Stat. 55(2), 777–790 (2019). MR3949953. https://doi.org/10.1214/18-aihp898

[26]

Coutin, L., Decreusefond, L.: Stein’s method for Brownian approximations. Commun. Stoch. Anal. 7(3), 349–372 (2014). MR3167403. https://doi.org/10.31390/cosa.7.3.01

[27]

Coutin, L., Decreusefond, L.: Higher order approximations via Stein’s method. Commun. Stoch. Anal. 8(2), 155–168 (2014). MR3269842. https://doi.org/10.31390/cosa.8.2.02

[28]

Coutin, L., Decreusefond, L.: Stein’s method for rough paths. Potential Anal. 53(2), 387–406 (2020). MR4125096. https://doi.org/10.1007/s11118-019-09773-z

[29]

Coutin, L., Decreusefond, L.: Donsker’s theorem in Wasserstein-1 distance. Electron. Commun. Probab. 25, 27 (2020), 13 pp. MR4089734. https://doi.org/10.1214/20-ecp308

[30]

Decreusefond, L.: The Stein-Dirichlet-Malliavin method. ESAIM Proc. Surveys, vol. 51. EDP Sci., Les Ulis (2015). MR3440790. https://doi.org/10.1051/proc/201551003

[31]

Decreusefond, L., Halconruy, H.: Malliavin and Dirichlet structures for independent random variables. Stoch. Process. Appl. 129(8), 2611–2653 (2019). MR3980139. https://doi.org/10.1016/j.spa.2018.07.019

[32]

Decreusefond, L., Schulte, M., Thäle, Ch.: Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry. Ann. Probab. 44(3), 2147–2197 (2016). MR3502603. https://doi.org/10.1214/15-AOP1020

[33]

Döbler, C.: Normal approximation via non-linear exchangeable pairs (2020). arXiv:2008.02272

[34]

Döbler, C., Kasprzak, M.: Stein’s method of exchangeable pairs in multivariate functional approximations. Elec. J. Probab. (2021, to appear). MR4235479. https://doi.org/10.18287/2541-7525-2020-26-2-23-49

[35]

Döbler, C., Kasprzak, M., Peccati, G.: Functional convergence of sequential U-processes with size-dependent kernels, arXiv:2008.02272 (2019).

[36]

Döbler, C., Peccati, G.: The Gamma Stein equation and non-central de Jong theorems. Bernoulli 24(4B), 3384–3421 (2018). MR3788176. https://doi.org/10.3150/17-BEJ963

[37]

Döbler, C., Peccati, G.: The fourth moment theorem on the Poisson space. Ann. Probab. 46(4), 1878–1916 (2018). MR3813981. https://doi.org/10.1214/17-AOP1215

[38]

Döbler, C., Vidotto, A., Zheng, G.: Fourth moment theorems on The Poisson space in any dimension. Electron. J. Probab. 23, 36 (2018), 27 pp. MR3798246. https://doi.org/10.1214/18-EJP168

[39]

Eichelsbacher, P., Thäle, C.: Malliavin-Stein method for Variance-Gamma approximation on Wiener space. Electron. J. Probab. 20(123), 1–28 (2014). MR3425543. https://doi.org/10.1214/EJP.v20-4136

[40]

Fathi, M.: Stein kernels and moment maps. Ann. Probab. 47(4), 2172–2185 (2019). MR3980918. https://doi.org/10.1214/18-AOP1305

[41]

Gaunt, R.E.: Variance-Gamma approximation via Stein’s method. Electron. J. Probab. 19(38), 1–33 (2014)

[42]

Gaunt, R.E.: Wasserstein and Kolmogorov error bounds for variance-gamma approximation via Stein’s method I. J. Theor. Probab. 33, 465–505 (2020). https://doi.org/10.1007/s10959-018-0867-4

[43]

Gaunt, R.E.: Stein factors for variance-gamma approximation in the Wasserstein and Kolmogorov distances (2020). arXiv:2008.06088. MR4064309. https://doi.org/10.1007/s10959-018-0867-4

[44]

Gaunt, R.E., Mijoule, G., Swan, Y.: An algebra of Stein operators. J. Math. Anal. Appl. 469(1), 260–279 (2019). MR3857522. https://doi.org/10.1016/j.jmaa.2018.09.015

[45]

Kasprzak, M.: Stein’s method for multivariate Brownian approximations of sums under dependence. Stoch. Process. Appl. 130(8), 4927–4967 (2020). MR4108478. https://doi.org/10.1016/j.spa.2020.02.006

[46]

Kesten, H., Lee, S.: The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6(2), 495–527 (1996). MR1398055. https://doi.org/10.1214/aoap/1034968141

[47]

Krein, Ch.: Weak convergence on Wiener space: targeting the first two chaoses. ALEA Lat. Am. J. Probab. Math. Stat. 16(1), 85–139 (2019). MR3903026. https://doi.org/10.30757/ALEA.v16-05

[48]

Lachièze-Rey, R., Peccati, G., Yang, X.: Quantitative two-scale stabilization on the Poisson space (2020). arXiv:2010.13362. MR4108865. https://doi.org/10.1090/proc/14964

[49]

Lachièze-Rey, R., Schultei, M., Yukich, J.: Normal approximation for stabilizing functionals. Ann. Appl. Probab. 29(2), 931–993 (2019). MR3910021. https://doi.org/10.1214/18-AAP1405

[50]

Last, G.: Stochastic analysis for Poisson processes. In: Stochastic Analysis for Poisson Point Processes. Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry, pp. 1–36. Springer (2016). MR3585396. https://doi.org/10.1007/978-3-319-05233-5_1

[51]

Last, G., Penrose, M.: Lectures on the Poisson Process. Cambridge University Press (2017). MR3791470

[52]

Ledoux, M.: Chaos of a Markov operator and the fourth moment condition. Ann. Probab. 40(6), 2439–2459 (2012). MR3050508. https://doi.org/10.1214/11-AOP685

[53]

Ledoux, M., Nourdin, I., Peccati, G.: Stein’s method, logarithmic Sobolev and transport inequalities. Geom. Funct. Anal. 25, 256–306 (2015). MR3320893. https://doi.org/10.1007/s00039-015-0312-0

[54]

Ledoux, M., Nourdin, I., Peccati, G.: A Stein deficit for the logarithmic Sobolev inequality. Sci. China Math. 60(7), 1163–1180 (2016). MR3665794. https://doi.org/10.1007/s11425-016-0134-7

[55]

Last, G., Peccati, G., Schulte, M.: Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Relat. Fields 165(3–4), 667–723 (2016). MR3520016. https://doi.org/10.1007/s00440-015-0643-7

[56]

Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Proceedings of the International Symposium on Stochastic Differential Equations, Kyoto, pp. 195–263. Wiley, New York (1976). 1978. MR0536013

[57]

Malliavin, P.: Stochastic Analysis. Springer (1997). MR1450093. https://doi.org/10.1007/978-3-642-15074-6

[58]

Marinucci, D., Rossi, M., Vidotto, A.: Non-universal fluctuations of the empirical measure for isotropic stationary fields on ${S^{2}}\times R$. Ann. App. Probab. (2021, to appear).

[59]

Notarnicola, M.: Fluctuations of nodal sets on the 3-torus and general cancellation phenomena (2020). arXiv:2004.04990

[60]

Nourdin, I.: In: Lectures on Gaussian approximations with Malliavin calculus. Sém. Probab., XLV, pp. 3–89 (2012). MR3185909. https://doi.org/10.1017/CBO9781139084659

[61]

Nourdin, I., Nualart, D.: The functional Breuer-Major theorem. Probab. Theory Relat. Fields 176, 203–218 (2020). MR4055189. https://doi.org/10.1007/s00440-019-00917-1

[62]

Nourdin, I., Nualart, D., Peccati, G.: The Breuer-Major theorem in total variation: improved rates under minimal regularity. Stoch. Process. Appl. 131, 1–20 (2021). MR4151212. https://doi.org/10.1016/j.spa.2020.08.007

[63]

Nourdin, I., Peccati, G.: Noncentral convergence of multiple integrals. Ann. Probab. 37(4), 1412–1426 (2009). MR2546749. https://doi.org/10.1214/08-AOP435

[64]

Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145(1–2), 75–118 (2009). MR2520122. https://doi.org/10.1007/s00440-008-0162-x

[65]

Nourdin, I., Peccati, G.: Cumulants on the Wiener space. J. Funct. Anal. 258(11), 3775–3791 (2010). MR2606872. https://doi.org/10.1016/j.jfa.2009.10.024

[66]

Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics. Cambridge University Press (2012). MR2962301. https://doi.org/10.1017/CBO9781139084659

[67]

Nourdin, I., Peccati, G.: The optimal fourth moment theorem. Proc. Am. Math. Soc. 143(7), 3123–3133 (2015). MR3336636. https://doi.org/10.1090/S0002-9939-2015-12417-3

[68]

Nourdin, I., Peccati, G., Rossi, M.: Nodal statistics of planar random waves. Commun. Math. Phys. 369(1), 99–151 (2019). MR3959555. https://doi.org/10.1007/s00220-019-03432-5

[69]

Nourdin, I., Poly, G.: Convergence in law in the second Wiener/Wigner chaos. Electron. Commun. Probab. 17(36), 1–12 (2012). MR2970700. https://doi.org/10.1214/ecp.v17-2023

[70]

Nourdin, I., Peccati, G., Reinert, G.: Second order Poincaré inequalities and CLTs on Wiener space. J. Funct. Anal. 257(4), 1005–1041 (2009). MR2527030. https://doi.org/10.1016/j.jfa.2008.12.017

[71]

Nourdin, I., Peccati, G., Reinert, G.: Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab. 38(5), 1947–1985 (2010). MR2722791. https://doi.org/10.1214/10-AOP531

[72]

Nourdin, I., Peccati, G., Swan, Y.: Entropy and the fourth moment phenomenon. J. Funct. Anal. 266(5), 3170–3207 (2014). MR3158721. https://doi.org/10.1016/j.jfa.2013.09.017

[73]

Nourdin, I., Peccati, G., Swan, Y.: Integration by parts and representation of information functionals. In: Proceedings of the 2014 IEEE International Symposium on Information Theory (ISIT), Honolulu, HI, pp. 2217–2221 (2015)

[74]

Nourdin, I., Peccati, G., Yang, X.: Berry-Esseen bounds in the Breuer-Major CLT and Gebelein’s inequality. Electron. Commun. Probab. 24, 34 (2019). MR3978683. https://doi.org/10.1214/19-ECP241

[75]

Nourdin, I., Peccati, G., Yang, X.:. Multivariate normal approximation on the Wiener space: new bounds in the convex distance (2020). arxiv:2001.02188. MR2606872. https://doi.org/10.1016/j.jfa.2009.10.024

[76]

Nourdin, I., Rosinski, J.: Asymptotic independence of multiple Wiener-Ito integrals and the resulting limit laws. Ann. Probab. 42(2), 497–526 (2014). MR3178465. https://doi.org/10.1214/12-AOP826

[77]

Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications, 2 edn. Springer, Berlin and Heidelberg and New York (2006). MR2200233

[78]

Nualart, D.: Malliavin Calculus and Its Applications. CBMS Regional Conference Series in Mathematics, vol. 110 (2009). MR2498953. https://doi.org/10.1090/cbms/110

[79]

Nualart, D., Ortiz-Latorre, S.: Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stoch. Process. Appl. 118(4), 614–628 (2008). MR2394845. https://doi.org/10.1016/j.spa.2007.05.004

[80]

Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005). MR2118863. https://doi.org/10.1214/009117904000000621

[81]

Peccati, G., Reitzner, M. (eds.): Stochastic Analysis for Poisson Point Processes. Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry Springer (2016). MR3444831. https://doi.org/10.1007/978-3-319-05233-5

[82]

Peccati, G., Rossi, M.: Quantitative limit theorems for local functionals of arithmetic random waves. In: Combinatorics in Dynamics, Stochastics and Control, The Abel Symposium, Rosendal, Norway, August 2016, vol. 13, pp. 659–689 (2018). MR3967400. https://doi.org/10.1007/978-3-030-01593-0_23

[83]

Peccati, G., Solé, J-., Utzet, F., Taqqu, M.S.: Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38, 443–478 (2010). MR2642882. https://doi.org/10.1214/10-AOP531

[84]

Peccati, G., Taqqu, M.S.: Wiener Chaos: Moments, Cumulants and Diagrams. Springer (2010). MR2791919. https://doi.org/10.1007/978-88-470-1679-8

[85]

Peccati, G., Vidotto, A.: Gaussian random measures generated by Berry’s nodal sets. J. Stat. Phys. 178(4), 443–478 (2020). MR4064212. https://doi.org/10.1007/s10955-019-02477-z

[86]

Penrose, M.D.: Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33(5), 1945–1991 (2005). MR2165584. https://doi.org/10.1214/009117905000000206

[87]

Penrose, M.D., Yukich, J.E.: Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11(4), 1005–1041 (2001). MR1878288. https://doi.org/10.1214/aoap/1015345393

[88]

Rossi, M.: Random nodal lengths and Wiener chaos. In: Probabilistic Methods in Geometry, Topology and Spectral Theory. Probabilistic Methods in Geometry, Topology and Spectral Theory. Contemporary Mathematics Series, vol. 739, pp. 155–169 (2019). MR4033918. https://doi.org/10.1090/conm/739/14898

[89]

Saumard, A.: Weighted Poincaré inequalities, concentration inequalities and tail bounds related to the behavior of the Stein kernel in dimension one. Bernoulli 25(4B), 3978–4006 (2019). MR4010979. https://doi.org/10.3150/19-BEJ1117

[90]

Schulte, M., Yukich, J.E.: Multivariate second order Poincaré inequalities for Poisson functionals. Electron. J. Probab. 24, 130 (2019), 42 pp. MR4040990. https://doi.org/10.1214/19-ejp386

[91]

Shih, H.-H.: On Stein’s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces. J. Funct. Anal. 261(5), 1236–1283 (2011). MR2807099. https://doi.org/10.1016/j.jfa.2011.04.016

[92]

Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, II, pp. 583–602 (1972). MR0402873

[93]

Stein, C.: Approximate computation of expectations. In: IMS. Lecture Notes-Monograph Series, vol. 7 (1986). MR0882007

[94]

Todino, A.P.: A quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of ${S^{2}}$. J. Math. Phys. 60(2), 023505 (2019). MR3916834. https://doi.org/10.1063/1.5048976

[95]

Trinh, K.D.: On central limit theorems in stochastic geometry for add-one cost stabilizing functionals. Electron. Commun. Probab. 24, 76 (2019), 15 pp. MR4049088. https://doi.org/10.1214/19-ecp279

[96]

Vidotto, A.: An improved second order Poincaré inequality for functionals of Gaussian fields (2017). arXiv:1706.06985. MR4064306. https://doi.org/10.1007/s10959-019-00883-3

[97]

Villani, C.: Optimal Transport. Old and New. Springer (2009). MR2459454. https://doi.org/10.1007/978-3-540-71050-9

[98]

Yogeshwaran, D., Subag, E., Adler, R.J.: Random geometric complexes in the thermodynamic regime. Probab. Theory Relat. Fields 167(1–2), 107–142 (2017). MR3602843. https://doi.org/10.1007/s00440-015-0678-9

Full article Related articles Cited by

Open access article under the CC BY license.

Keywords

Limit theorems Stein’s method Malliavin calculus Wiener space Poisson space multiple integral Markov triple Markov generator eigenspace eigenfunction spectrum functional Γ-calculus weak convergence fourth moment theorems Berry–Essen bounds probability metrics 60F05 60B10 28C20 60H07 47D07 34L10 47A10

Funding

Giovanni Peccati is supported by the FNR grant FoRGES (GS1R-AGR-3376-10) at Luxembourg University. Xiaochuan Yang is supported by the EPSRC grant EP/T028653/1.

Metrics

since March 2018

946 Article info views	980 Full article views
604 PDF downloads	60 XML downloads

RSS

Authors

Abstract

References

Export citation

Copy and paste formatted citation

Download citation in file