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arXiv · 1705.10485

Mod-$ϕ$ convergence, II: Estimates on the speed of convergence

Valentin Féray·Pierre-Loïc Méliot·Ashkan Nikeghbali
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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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Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali. 2018-02-20. Mod-$ϕ$ convergence, II: Estimates on the speed of convergence. https://arxiv.org/abs/1705.10485

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