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The asymptotic behavior of expressions of the form $%\sum_{t=1}^{n}f(r_{n}x_{t})$ where $x_{t}$ is an integrated process, $r_{n}$ isa sequence of norming constants, and $f$ is a measurable function has been thesubject of a number of articles in recent years. We mention Borodin andIbragimov (1995), Park and Phillips (1999), de Jong (2004), Jeganathan (2004),P\"{o}tscher (2004), de Jong and Whang (2005), Berkes and Horvath (2006), andChristopeit (2009) which study weak convergence results for such expressionsunder various conditions on $x_{t}$ and the function $f$. Of course, theseresults 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 resultis available for the case where $f$ is non-integrable with respect toLebesgue-measure in a neighborhood of a given point, say $x=0$. In this paperwe are interested in bounds on the order of magnitude of $%\sum_{t=1}^{n}|x_{t}| ^{-\alpha}$ when $\alpha \geq 1$, a case where theimplied function $f$ is not integrable in any neighborhood of zero. Moregenerally, 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 variablessatisfying certain conditions.

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