# Mod-$\phi$ convergence, II: Estimates on the speed of convergence

## TL;DR

This paper develops estimates for the rate of convergence to stable laws within the mod-$$ convergence framework, introducing a zone of control concept and applying it to various probabilistic models.

## Contribution

It introduces a notion of zone of control for mod-$$ convergence and derives Berry-Esseen type estimates, broadening the understanding of convergence speeds in probability theory.

## Key findings

- Estimates of convergence speed for stable laws in various models
- Application to Brownian motion, random matrices, and Ising model
- Recovery of known results using new mod-$$ convergence techniques

## Abstract

In this paper, we give estimates for the speed of convergence towards a limiting stable law in the recently introduced setting of mod-$\phi$ convergence. Namely, we define a notion of zone of control, closely related to mod-$\phi$ 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.