Published online by Cambridge University Press: 12 November 2012
Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.
I thank Kalidas Jana for inquiring about the order of magnitude of some of the quantities now treated in the paper. I am indebted to Robert de Jong for comments on an early draft that have led to an improvement in Theorem 1. I am grateful to Istvan Berkes, Hannes Leeb, David Preinerstorfer, Zhan Shi, the referees, and the editor Peter Phillips for helpful comments.