  An exchangeable pair approach is commonly taken in the normal and non-normal
approximation using Stein's method. It has been successfully used to identify
the limiting distribution and provide an error of approximation. However, when
the difference of the exchangeable pair is not bounded by a small deterministic
constant, the error bound is often not optimal. In this paper, using the
exchangeable pair approach of Stein's method, a new Berry-Esseen bound for an
arbitrary random variable is established without a bound on the difference of
the exchangeable pair. An optimal convergence rate for normal and non-normal
approximation is achieved when the result is applied to various examples
including the quadratic forms, general Curie-Weiss model, mean field Heisenberg
model and colored graph model.
