Rate of Convergence for Sums and Maxima and Doubly Ideal Metrics

It is well known that the minimal $L_p$-metrics are ideal with respect to summation and maxima of order $r=\min(p,1)$. This implies that one can get rate of convergence results in stable limit theorems with $0<\alpha<1$ with respect to maxima and sums. It will be shown that one can extend and improve the ideality properties of minimal $L_p$-metrics to stable limit theorems with $0<\alpha<2$. As a consequence one obtains, e.g., an improvement of the classical results on the rate of convergence of sums with values in Banach spaces with respect to the Prokhorov distance. In the second part of the paper it is proved that a problem posed by Zolotarev in 1983 on the existence of doubly ideal metrics of order $r>1$ has an essential negative answer. In spite of this the minimal $L_p$-metrics behave like ideal metrics of order $r>1$ with respect to maxima and sums. This allows to improve results on the stability of queueing models respect to departures from the ideal model.