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

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