  Mathematical mean-field approaches play an important role in different fields
of Physics and Chemistry, but have found in recent works also their application
in Economics, Finance and Game Theory. The objective of our paper is to
investigate a special mean-field problem in a purely stochastic approach: for
the solution $(Y,Z)$ of a mean-field backward stochastic differential equation
driven by a forward stochastic differential of McKean--Vlasov type with
solution $X$ we study a special approximation by the solution $(X^N,Y^N,Z^N)$
of some decoupled forward--backward equation which coefficients are governed by
$N$ independent copies of $(X^N,Y^N,Z^N)$. We show that the convergence speed
of this approximation is of order $1/\sqrt{N}$. Moreover, our special choice of
the approximation allows to characterize the limit behavior of
$\sqrt{N}(X^N-X,Y^N-Y,Z^N-Z)$. We prove that this triplet converges in law to
the solution of some forward--backward stochastic differential equation of
mean-field type, which is not only governed by a Brownian motion but also by an
independent Gaussian field.
