  The first aim of this paper is to establish the weak convergence rate of
nonlinear two-time-scale stochastic approximation algorithms. Its second aim is
to introduce the averaging principle in the context of two-time-scale
stochastic approximation algorithms. We first define the notion of asymptotic
efficiency in this framework, then introduce the averaged two-time-scale
stochastic approximation algorithm, and finally establish its weak convergence
rate. We show, in particular, that both components of the averaged
two-time-scale stochastic approximation algorithm simultaneously converge at
the optimal rate $\sqrt{n}$.
