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

TL;DR

This paper investigates the asymptotic order of sums involving negative powers of integrated processes, especially when the function is non-integrable near zero, extending existing results to this challenging case.

Contribution

It provides new bounds on the magnitude of sums of negative powers of integrated processes when the function is non-integrable near zero, a case not previously addressed.

Findings

Bounds on the order of sums of |x_t|^{-α} for α ≥ 1

Extension of asymptotic results to non-integrable functions near zero

Inclusion of random weights v_t in the analysis

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.