A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

Soren Johansen and Morten Nielsen

No 10-29, Discussion Papers from University of Copenhagen. Department of Economics

Abstract: We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=Δ^(-d)u(t), where d є (-1/2,1/2) is the fractional integration parameter and u(t) is weakly dependent. The classical condition is existence of q>max(2,(d+1/2)-¹) moments of the innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that under some relatively weak conditions on u(t), the existence of q≥max(2,(d+1/2)-¹) is in fact necessary for the FCLT for fractionally integrated processes and that q>max(2,(d+1/2)-¹) moments are necessary and sufficient for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed, which is remarkable because it is the only FCLT where the moment condition has been weakened relative to the earlier condition. As a corollary to our main theorem we show that their moment condition is not sufficient.

Keywords: fractional integration; functional central limit theorem; long memory; moment condition; necessary condition (search for similar items in EconPapers)
JEL-codes: C22 (search for similar items in EconPapers)
Pages: 8 pages
Date: 2010-10
References: View complete reference list from CitEc
Citations: View citations in EconPapers (5)

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Related works:
Journal Article: A NECESSARY MOMENT CONDITION FOR THE FRACTIONAL FUNCTIONAL CENTRAL LIMIT THEOREM (2012)
Working Paper: A necessary moment condition for the fractional functional central limit theorem (2010)
Working Paper: A Necessary Moment Condition For The Fractional Functional Central Limit Theorem (2010)
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