On the Rate of Multivariate Poisson Convergence

The distribution of the sum of independent nonidentically distributed Bernoulli random vectors inRkis approximated by a multivariate Poisson distribution. By using a multivariate adaption of Kerstanʹs (1964,Z. Wahrsch. verw. Gebiete2, 173–179) method, we prove a conjecture of Barbour (1988,J. Appl. Probab.25A, 175–184) on removing a log-term in the upper bound of the total variation distance. Second-order approximations are included.