  In this paper we use a Malliavin-Stein type method to investigate Poisson and
normal approximations for the measurable functions of infinitely many
independent random variables. We combine Stein's method with the difference
operators in theory of concentration inequalities to obtain explicit bounds on
Wasserstein, Kolmogorov and total variation distances. When restricted to the
functions of a finite number of independent random variables, our method
provides new bounds in the normal approximation. Meanwhile, our bounds in
Poisson approximation are first to obtain explicitly.
