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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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Correspondence to Ashkan Nikeghbali .

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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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