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

  • Valentin Féray
  • Pierre-Loïc Méliot
  • Ashkan NikeghbaliEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2252)


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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Valentin Féray
    • 1
  • Pierre-Loïc Méliot
    • 2
  • Ashkan Nikeghbali
    • 1
    Email author
  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland
  2. 2.Université Paris-Sud - Faculté des Sciences d’OrsayInstitut de mathématiques d’OrsayOrsayFrance

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