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On the Order of Magnitude of Sums of Negative Powers of Integrated Processes

The asymptotic behavior of expressions of the form $% \sum_{t=1}^{n}f(r_{n}x_{t})$ where $x_{t}$ is an integrated process, $r_{n}$ is a sequence of norming constants, and $f$ is a measurable function has been the subject of a number of articles in recent years. We mention Borodin and Ibragimov (1995), Park and Phillips (1999), de Jong (2004), Jeganathan (2004), Pötscher (2004), de Jong and Whang (2005), Berkes and Horvath (2006), and Christopeit (2009) which study weak convergence results for such expressions under various conditions on $x_{t}$ and the function $f$. Of course, these results also provide information on the order of magnitude of $% \sum_{t=1}^{n}f(r_{n}x_{t})$. However, to the best of our knowledge no result is available for the case where $f$ is non-integrable with respect to Lebesgue-measure in a neighborhood of a given point, say $x=0$. In this paper we are interested in bounds on the order of magnitude of $% \sum_{t=1}^{n}|x_{t}| ^{-α}$ when $α\geq 1$, a case where the implied function $f$ is not integrable in any neighborhood of zero. More generally, we shall also obtain bounds on the order of magnitude for $\sum_{t=1}^{n}v_{t}|x_{t}| ^{-α}$ where $v_{t}$ are random variables satisfying certain conditions.