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

Abstract

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), P\"{o}tscher (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 $% \sum_{t=1}^{n}|x_{t}| ^{-\alpha}$ when $\alpha \geq 1$, 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 $\sum_{t=1}^{n}v_{t}|x_{t}| ^{-\alpha}$ where $v_{t}$ are random variables satisfying certain conditions.