A new metric is introduced which is suitable for estimating the rate of convergence of processes related to stable random variables. It is shown that it has an upper bound depending on the difference pseudomoments, but not on the absolute moments. This new metric is then applied to get some rates of convergence to a self-similar process constructed from a stable process.
"An Ideal Metric and the Rate of Convergence to a Self-Similar Process." Ann. Probab. 15 (2) 708 - 727, April, 1987. https://doi.org/10.1214/aop/1176992167