  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.
