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.