On Stein-Chen factors for Poisson approximation

The Stein-Chen method for the estimation of upper bounds of the distance of the whole distribution of a point process from that of a Poisson process was investigated in Barbour and Brown (1992). A feature of the Stein-Chen approximation for random variables is the existence of "magic" factors which decrease with the mean of the Poisson distribution. Using a Wasserstein metric for process approximation, one of these factors behaves in a similar way to that for random variables but the other involves an additional logarithm. Counterexamples are presented here to show that the logarithmic factor is necessary.