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Mod-ϕ Convergence, II: Estimates on the Speed of Convergence

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Séminaire de Probabilités L

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2252))

Abstract

In this paper, we give estimates for the speed of convergence towards a limiting stable law in the recently introduced setting of mod-ϕ convergence. Namely, we define a notion of zone of control, closely related to mod-ϕ convergence, and we prove estimates of Berry–Esseen type under this hypothesis. Applications include:

  • the winding number of a planar Brownian motion;

  • classical approximations of stable laws by compound Poisson laws;

  • examples stemming from determinantal point processes (characteristic polynomials of random matrices and zeroes of random analytic functions);

  • sums of variables with an underlying dependency graph (for which we recover a result of Rinott, obtained by Stein’s method);

  • the magnetization in the d-dimensional Ising model;

  • and functionals of Markov chains.

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References

  1. N. Alon, J. Spencer, The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization, 3rd edn. (Wiley, New York, 2008)

    Google Scholar 

  2. O. Arizmendi, T. Hasebe, F. Lehner, C. Vargas, Relations between cumulants in noncommutative probability. Adv. Math. 282, 56–92 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Baldi, Y. Rinott, On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17(4), 1646–1650 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. A.D. Barbour, M. Karonśki, A. Rucinśki, A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory B 47(2), 125–145 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. A.D. Barbour, E. Kowalski, A. Nikeghbali, Mod-discrete expansions. Probab. Theory Relat. Fields 158(3), 859–893 (2009)

    MathSciNet  MATH  Google Scholar 

  6. A.C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49(1), 122–136 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Billingsley, Probability and Measure. Wiley Series in Probability and Mathematical Statistics, 3rd edn. (Wiley, New York, 1995)

    Google Scholar 

  8. E. Bolthausen, The Berry–Esseen theorem for functionals of discrete Markov chains. Z. Wahr. Verw. Geb. 54, 59–73 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Bóna, On three different notions of monotone subsequences, in Permutation Patterns. London Mathematical Society. Lecture Note Series, vol. 376 (Cambridge University Press, Cambridge, 2010), pp. 89–113

    Google Scholar 

  10. P. Bourgade, C. Hughes, A. Nikeghbali, M. Yor, The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J. 145, 45–69 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. A.V. Bulinskii, Rate of convergence in the central limit theorem for fields of associated random variables. Theory Probab. Appl. 40(1), 136–144 (1996)

    Article  Google Scholar 

  12. R. Chhaibi, F. Delbaen, P.-L. Méliot, A. Nikeghbali, Mod-ϕ convergence: approximation of discrete measures and harmonic analysis on the torus. arXiv:1511.03922 (2015)

    Google Scholar 

  13. R. Cogburn, The central limit theorem for Markov processes, in Proceedings of the Sixth Annual Berkeley Symposium on Mathematical Statistics and Probability, vol. 2 (University of California Press, Berkeley, 1972), pp. 485–512

    Google Scholar 

  14. F. Delbaen, E. Kowalski, A. Nikeghbali, Mod-ϕ convergence. Int. Math. Res. Not. 2015(11), 3445–3485 (2015)

    MathSciNet  MATH  Google Scholar 

  15. P. Diaconis, D. Stroock, Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1(1), 36–61 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Duneau, D. Iagolnitzer, B. Souillard, Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems. Commun. Math. Phys. 31(3), 191–208 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Duneau, D. Iagolnitzer, B. Souillard, Strong cluster properties for classical systems with finite range interaction. Commun. Math. Phys. 35, 307–320 (1974)

    Article  MathSciNet  Google Scholar 

  18. R. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer, New York, 1985)

    Book  MATH  Google Scholar 

  19. W. Feller, An Introduction to Probability Theory and Its Applications. Wiley Series in Probability and Mathematical Statistics, vol. II, 2nd edm. (Wiley, New York, 1971)

    Google Scholar 

  20. V. Féray, Weighted dependency graphs. arXiv preprint 1605.03836 (2016)

    Google Scholar 

  21. V. Féray, P.-L. Méliot, A. Nikeghbali, Mod-ϕconvergence: Normality Zones and Precise Deviations. Springer Briefs in Probability and Mathematical Statistics (Springer, Cham, 2016)

    Google Scholar 

  22. J.A. Fill, Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab. 1(1), 62–87 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Friedli, Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction (Cambridge University Press, Cambridge, 2017)

    Book  MATH  Google Scholar 

  24. L. Goldstein, N. Wiroonsri, Stein’s method for positively associated random variables with applications to Ising, percolation and voter models. Ann. Inst. Henri Poincaré 54, 385–421 (2018). Preprint arXiv:1603.05322

    Google Scholar 

  25. G. Grimmett, Weak convergence using higher-order cumulants. J. Theor. Probab. 5(4), 767–773 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. O. Häggström, On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Relat. Fields 132, 74–82 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Jacod, E. Kowalski, A. Nikeghbali, Mod-Gaussian convergence: new limit theorems in probability and number theory. Forum Math. 23(4), 835–873 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Janson, Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs. Ann. Probab. 16, 305–312 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. G.L. Jones, On the Markov chain central limit theorem. Probab. Surv. 1, 299–320 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Kato, Perturbation Theory for Linear Operators. Classics in Mathematics, 2nd edn. (Springer, Berlin, 1980)

    Google Scholar 

  31. J.P. Keating, N.C. Snaith, Random matrix theory and \(\zeta (\frac {1}{2}+\mathrm {i}t)\). Commun. Math. Phys. 214(1), 57–89 (2000)

    Google Scholar 

  32. J.G. Kemeny, J.L. Snell, Finite Markov Chains. Undergraduate Texts in Mathematics (Springer, Berlin, 1976)

    Google Scholar 

  33. C. Kipnis, S.R.S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104, 1–19 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. V.Y. Korolev, I.G. Shevtsova, On the upper bound for the absolute constant in the Berry–Esseen inequality. Theory Probab. Appl. 54(4), 638–658 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Kowalski, A. Nikeghbali, Mod-Poisson convergence in probability and number theory. Int. Math. Res. Not. 18, 3549–3587 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Kowalski, A. Nikeghbali, Mod-Gaussian distribution and the value distribution of \(\zeta (\frac {1}{2}+\mathrm {i}t)\) and related quantities. J. Lond. Math. Soc. 86(2), 291–319 (2012)

    Google Scholar 

  37. K. Krokowski, A. Reichenbachs, C. Thäle, Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs, point processes and percolation. arXiv:1503.01029v1 [math.PR] (2015)

    Google Scholar 

  38. S. Lang, Real and Functional Analysis. Graduate Texts in Mathematics, 3rd edn., vol. 142 (Springer, New York, 1993)

    Google Scholar 

  39. V.P. Leonov, A.N. Shiryaev, On a method of calculation of semi-invariants. Theory Prob. Appl. 4, 319–329 (1959)

    Article  MATH  Google Scholar 

  40. P. Lezaud, Chernoff-type bound for finite Markov chains, PhD thesis, Université Paul Sabatier, Toulouse, 1996

    Google Scholar 

  41. P. Malliavin, Integration and Probability. Graduate Texts in Mathematics, vol. 157 (Springer, New York, 1995)

    Google Scholar 

  42. V.A. Malyshev, R.A. Minlos, Gibbs Random Fields (Springer, Dordrecht, 1991)

    Book  MATH  Google Scholar 

  43. B. Mann, Berry–Esseen central limit theorems for Markov chains, PhD thesis, Harvard University, 1996

    Google Scholar 

  44. P.-L. Méliot, A. Nikeghbali, Mod-Gaussian convergence and its applications for models of statistical mechanics, in In Memoriam Marc Yor – Séminaire de Probabilités XLVII. Lecture Notes in Mathematics, vol. 2137 (Springer, Cham, 2015), pp. 369–425

    Google Scholar 

  45. C.M. Newman, Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74(2), 119–128 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Penrose, J. Yukich, Normal approximation in geometric probability, in Stein’s Method and Applications. Lecture Note Series, vol. 5 (Institute for Mathematical Sciences, National University of Singapore, Singapore, 2005), pp. 37–58

    Google Scholar 

  47. Y. Rinott, On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55(2), 135–143 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  48. W. Rudin, Functional Analysis, 2nd edn. (McGraw-Hill, New York, 1991)

    MATH  Google Scholar 

  49. L. Saloff-Coste, Lectures on finite Markov chains, in Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Mathematics, vol. 1665 (Springer, Berlin, 1997), pp. 301–413

    Google Scholar 

  50. K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68 (Cambridge University Press, Cambridge, 1999)

    Google Scholar 

  51. K.-I. Sato, M. Yamazato, Stationary processes of ornstein-uhlenbeck type, in Probability Theory and Mathematical Statistics, Fourth USSR-Japan Symposium, ed. by K. Itô, J.V. Prokhorov. Lecture Notes in Mathematics, vol. 1021 (Springer, Berlin, 1983), pp. 541–551

    Google Scholar 

  52. L. Saulis, V.A. Statulevičius, Limit Theorems for Large Deviations. Mathematics and Its Applications (Soviet Series), vol. 73 (Kluwer Academic Publishers, Dordrecht, 1991)

    Google Scholar 

  53. F. Spitzer, Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87, 187–197 (1958)

    MathSciNet  MATH  Google Scholar 

  54. V.A. Statulevičius, On large deviations. Probab. Theory Relat. Fields 6(2), 133–144 (1966)

    MATH  Google Scholar 

  55. D.W. Stroock, Probability Theory: An Analytic View, 2nd edn. (Cambridge University Press, Cambridge, 2011)

    MATH  Google Scholar 

  56. T. Tao, Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol. 132 (American Mathematical Society, Providence, 2012)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Valentin Féray & Ashkan Nikeghbali

  2. Université Paris-Sud - Faculté des Sciences d’Orsay, Institut de mathématiques d’Orsay, Orsay, France

    Pierre-Loïc Méliot

Authors
  1. Valentin Féray
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  2. Pierre-Loïc Méliot
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  3. Ashkan Nikeghbali
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Correspondence to Ashkan Nikeghbali .

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Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France

    Alain Rouault

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Féray, V., Méliot, PL., Nikeghbali, A. (2019). Mod-ϕ Convergence, II: Estimates on the Speed of Convergence. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités L. Lecture Notes in Mathematics(), vol 2252. Springer, Cham. https://doi.org/10.1007/978-3-030-28535-7_15

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