A Necessary Moment Condition For The Fractional Functional Central Limit Theorem
We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x = Δ − u , where null is the fractional integration parameter and u is weakly dependent. The classical condition is existence of q ≥ 2 and null moments of the innovation sequence. When d is close to null this moment condition is very strong. Our main result is to show that when null and under some relatively weak conditions on u , the existence of null moments is in fact necessary for the FCLT for fractionally integrated processes and that null moments are necessary for more general fractional processes. Davidson and de Jong (2000, Econometric Theory 16, 643–666) presented a fractional FCLT where only q > 2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient and hence that their result is incorrect.
Volume (Year): 28 (2012)
Issue (Month): 03 (June)
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- de Jong, Robert M. & Davidson, James, 2000.
"The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals I,"
Cambridge University Press, vol. 16(05), pages 621-642, October.
- Davidson, James & de Jong, Robert M., 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals Ii," Econometric Theory, Cambridge University Press, vol. 16(05), pages 643-666, October.
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