  We show by a surprisingly simple argument that the exchangeability condition,
which is key to the exchangeable pair approach in Stein's method for
distributional approximation, can be omitted in many standard settings. This is
achieved by replacing the usual antisymmetric function by a simpler one, for
which only equality in distribution is required. In the case of normal
approximation we also slightly improve the constants appearing in previous
results. For Poisson approximation, a different antisymmetric function is used,
and additional error terms are needed if the bound is to be extended beyond the
exchangeable setting.

Probability