• Regulated time series;
  • fractionally integrated time series;
  • fractional Brownian motion

Regulated (bounded) integrated time series are of significant practical importance and a recent development in the time series literature. Although regulated integrated series are characterized by asymptotic distributions that differ substantially from their unregulated counterparts, most inferential exercises continue to be performed with complete disregard for this potential feature of time series data. To date, only Cavaliere (2005) and Cavaliere and Xu (2011) have attempted to develop a theory for regulated integrated time series, particularly in the context of unit root testing. Unfortunately, no such theory has been developed for regulated fractionally integrated series, which are particularly important in financial time series and also in some unit root testing literature. This article achieves just this: it establishes a framework for regulated fractionally integrated processes and develops their functional central limit distributions. In addition, this article presents some simulation evidence and discusses several algorithms for obtaining the limiting distributions for these processes.