Integrated functionals of normal and fractional processes

Boris Buchmann, Ngai Hang Chan

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6 Citations (Scopus)


Consider Zft (u) = Rtu 0 f(Ns) ds, t > 0, u 2 [0, 1] where N = (Nt)t2R is a normal process and f is a measurable real-valued function satisfying Ef(N0)2 <1 and Ef(N0) = 0. If the dependence is sufficiently weak Hariz [16] showed that Zf t /t1/2 converges in distribution to a multiple of standard Brownian motion as t 1. If the dependence is sufficiently strong then Zt/(EZt(1)2)1/2 converges in distribution to a higher order Hermite process as t 1 by a result by Taqqu [27]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in details and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices H 2 (0, 1), we give their limiting distributions. In this context, we show that the known results are only applicable to H <3/4 and H > 3/4, respectively, whereas our result covers H = 3/4.
Original languageEnglish
Pages (from-to)49 - 70
Number of pages22
JournalAnnals of Applied Probability
Issue number1
Publication statusPublished - 2009

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