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


Féray, Valentin; Méliot, Pierre-Loïc; Nikeghbali, Ashkan (2019). Mod-ϕ Convergence, II: Estimates on the Speed of Convergence. In: Donati-Martin, Catherine; Lejay, Antoine; Rouault, Alain. Séminaire de Probabilités, L. Cham, Switzreland: Springer, 405-477.

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.

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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Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Language:English
Date:1 January 2019
Deposited On:08 Jan 2020 14:18
Last Modified:08 Jan 2020 14:18
Publisher:Springer
Series Name:Lecture Notes in Mathematics
ISSN:0075-8434
ISBN:9783030285340
OA Status:Green
Publisher DOI:https://doi.org/10.1007/978-3-030-28535-7_15

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