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-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.