Electronic Journal of Probability

On Stein’s method for multivariate self-decomposable laws with finite first moment

Benjamin Arras and Christian Houdré

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We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R} ^d$ having finite first moment. Building on previous univariate findings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy specifically designed for infinitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\mathbb{R} ^d$ and a non-degenerate self-decomposable target law with finite second moment. Finally, under an appropriate Poincaré-type inequality assumption, we investigate, via variational methods, the existence of Stein kernels. In particular, this leads to quantitative versions of classical results on characterizations of probability distributions by variational functionals.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 29, 33 pp.

Received: 4 October 2018
Accepted: 2 March 2019
First available in Project Euclid: 26 March 2019

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Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms 60F05: Central limit and other weak theorems

infinite divisibility self-decomposability Stein’s method Stein’s kernel weak limit theorems rates of convergence smooth Wassertein distance

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Arras, Benjamin; Houdré, Christian. On Stein’s method for multivariate self-decomposable laws with finite first moment. Electron. J. Probab. 24 (2019), paper no. 29, 33 pp. doi:10.1214/19-EJP285. https://projecteuclid.org/euclid.ejp/1553565780

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  • [1] B. Arras and C. Houdré. On Stein’s method for infinitely divisible laws with finite first moment. To appear: SpringerBriefs in Probability and Mathematical Statistics, arXiv:1712.10051, 2019.
  • [2] A. D. Barbour. Stein’s method for diffusion approximations. Probab. Theory Relat. Fields. 84(3), 297-322, 1990.
  • [3] D. Bakry, I. Gentil and M. Ledoux. Analysis and Geometry of Markov diffusion operators. Springer, 2014.
  • [4] V. I. Bogachev. Measure Theory Springer, Vol. 1, 2007.
  • [5] L. Bondesson. Generalized Gamma Convolutions and Related Classes of Distributions and Densities Lectures Notes in Statistics, Springer, 1992.
  • [6] A.A. Borovkov and S.A. Utev. On an inequality and a related characterisation of the normal distribution. Theory Probab. Appl., 28:219–228, 1984.
  • [7] T. Cacoullos, V. Papathanasiou and S. Utev. Variational inequalities with examples and an application to the central limit theorem. Ann. Probab., 22:1607–1618, 1994.
  • [8] S. Chatterjee. A new method of normal approximation. Ann. Probab. 36(4):1584–1610, 2008.
  • [9] S. Chatterjee. Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143:1–40, 2007.
  • [10] S. Chatterjee. A new approach to strong embeddings. Probab. Theory Related Fields 152:231–264, 2012.
  • [11] S. Chatterjee and E. Meckes. Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4, 257–283, 2008.
  • [12] L.H.Y. Chen. Poincaré-type inequalities via stochastic integrals. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 69:251–277, 1985.
  • [13] L.H.Y. Chen, L. Goldstein, and Q.M. Shao. Normal Approximation by Stein’s Method Probability and its Application, Springer, Heidelberg, 2011.
  • [14] L. H. Y. Chen and J. H. Lou. Characterization of probability distributions by Poincaré-type inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 23(1): 91–110, 1987.
  • [15] Z.-Q. Chen and X. Zhang. Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Relat. Fields 165:267–312, 2016.
  • [16] T. A. Courtade, M. Fathi and A. Pananjady. Existence of Stein kernels under a spectral gap, and discrepancy bounds. To appear: Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1703.07707, 2018.
  • [17] L. C. Evans. Partial Differential Equations Graduate Studies in Mathematics, Vol. 19, 2002.
  • [18] M. Fathi. Stein kernel and moment maps. To appear: Ann. Probab., arXiv:1804.04699, 2018.
  • [19] B. V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Translated from the Russian, annotated and revised by K.-L. Chung, with appendices by J. L. Doob and P. L. Hsu. Addison-Wesley, revised edition, 1968.
  • [20] L. Goldstein and Y. Rinott. Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33(1):1–17, 1996.
  • [21] J. Gorham, A. B. Duncan, S.J. Vollmer and L. Mackey. Measuring sample quality with diffusions. To appear: Ann. Appl. Probab. arXiv:1611.06972, 2016.
  • [22] F. Götze. On the rate of convergence in the multivariate CLT. Ann. Probab. 19(2):724–739, 1991.
  • [23] C. Houdré, V. Pérez-Abreu and D. Surgailis. Interpolation, correlation identities and inequalities for infinitely divisible variables. J. Fourier Anal. Appl. 4(6):651–668, 1998.
  • [24] A. Ya. Khintchine. Limit Laws for Sums of Independent Random Variables. ONTI, Moscow–Leningrad (in Russian), 1938.
  • [25] M. Ledoux, I. Nourdin and G. Peccati. Stein’s method, logarithmic Sobolev and transport inequalities. Geom. Funct. Anal., 25:256–306, 2015.
  • [26] P. Lévy. Théorie de l’Addition des Variables Aléatoires 2nd, Gauthier-Villars, Paris, (1st ed. 1937), 1954.
  • [27] C. Ley, G. Reinert and Y. Swan. Stein’s method for comparison of univariate distributions. Probab. Surveys, 14:1–52, 2017.
  • [28] M. Loève. Probability Theory, I and II Springer, New-York. (1st ed., Van Nostrand, Princeton, NJ, 1955), 1977,1978.
  • [29] L. Mackey and J. Gorham. Multivariate Stein factors for a class of strongly log-concave distributions. Electron. Commun. Probab., 21(56):1–14, 2016.
  • [30] M. B. Marcus and J. Rosinski. $L^1$-norm of infinitely divisible random vectors and certain stochastic integrals. Electron. Commun. Probab. 6:15–29, 2001.
  • [31] E. Meckes. On Stein’s method for multivariate normal approximation. High Dimensional Probability V: The Luminy Volume, Institute of Mathematical Statistics, 153–178, 2009.
  • [32] I. Nourdin and G. Peccati. Stein’s method on Wiener chaos. Probab. Theory Related Fields, 145(1):75–118, 2008.
  • [33] I. Nourdin and G. Peccati. Normal Approximations with Malliavin calculus, volume 192 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2012. From Stein’s method to universality.
  • [34] I. Nourdin, G. Peccati and A. Réveillac. Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Statist., 46(1):45–58, 2010.
  • [35] I. Nourdin, G. Peccati and Y. Swan. Entropy and the fourth moment phenomenon. J. Funct. Anal., 266:3170–3207, 2014.
  • [36] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark. NIST Handbook of Mathematical Functions Cambridge University Press, 2010.
  • [37] V. Pérez-Abreu and R. Stelzer. Infinitely divisible multivariate and matrix gamma distributions. J. Multivariate Anal., 130:155–175, 2014.
  • [38] V.V. Petrov Limit Theorems of Probability Theory. Oxford University Press, Oxford, 1995.
  • [39] M. Raic. A multivariate CLT for decomposable random vectors with finite second moments. J. Theoret. Probab. 17(3), 573–603, 2004.
  • [40] G. Reinert and A. Röllin. Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab., 37(6):2150–2173, 2009.
  • [41] Y. Rinott and V. Rotar. A multivariate CLT for local dependence with $n^{-1/2}\log (n)$ rate and applications to multivariate graph related statistics. J. Multivariate Anal., 56:333–350, 1996.
  • [42] A. Röllin. Stein’s method in high dimensions with applications. Ann. Inst. Henri Poincaré Probab. Stat., 49(2):529–549, 2013.
  • [43] N.F. Ross Fundamentals of Stein’s method. Probab. Surv. 8:210–293, 2011.
  • [44] K-I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Corrected Printing with Supplements, 2015.
  • [45] C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 583–602, 1972.
  • [46] C. Stein. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes Monograph Series, 7. Institute of Mathematical Statistics, 1986.
  • [47] F.W. Steutel and K. Van Harn. Infinite Divisibility of Probability Distributions on the Real Line CRC Press, 2003.
  • [48] K. Takano. The Lévy representation of the characteristic function of the probability density $Ce^{-\|x\|}dx$. Bull. Fac. Sci., Ibaraki Univ. 20:61–65,1988.
  • [49] K. Takano. The Lévy representation of the characteristic function of the probability density $\Gamma (m+d/2)\left (\pi ^2\Gamma (m)\right )^{-1}(1+|x|^2)^{-m-d/2}$. Bull. Fac. Sci., Ibaraki Univ. 21:21–27,1989.
  • [50] K. Takano. On mixtures of the normal distribution by the generalized gamma convolutions. Bull. Fac. Sci., Ibaraki Univ. 21:29–41,1989.
  • [51] S.A. Utev. Probability problems connected with a certain integrodifferential inequality. Sib. Math. J. 30(3):490–493, 1989.