The Annals of Applied Probability

Integrated functionals of normal and fractional processes

Boris Buchmann and Ngai Hang Chan

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Consider Ztf(u)=0tuf(Ns) ds, t>0, u∈[0, 1], where N=(Nt)t∈ℝ is a normal process and f is a measurable real-valued function satisfying Ef(N0)2<∞ and Ef(N0)=0. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191–216] showed that Ztf/t1/2 converges in distribution to a multiple of standard Brownian motion as t→∞. If the dependence is sufficiently strong, then Zt/(EZt(1)2)1/2 converges in distribution to a higher order Hermite process as t→∞ by a result by Taqqu [Wahrsch. Verw. Gebiete 50 (1979) 53–83]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in detail 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∈(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.

Article information

Ann. Appl. Probab., Volume 19, Number 1 (2009), 49-70.

First available in Project Euclid: 20 February 2009

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 60G15: Gaussian processes 60J65: Brownian motion [See also 58J65] 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators

Brownian motion fractional Brownian motion fractional Ornstein–Uhlenbeck process Gaussian processes Hermite process noncentral and central functional limit theorems nonstandard scaling Rosenblatt process slowly varying norming unit root problem


Buchmann, Boris; Chan, Ngai Hang. Integrated functionals of normal and fractional processes. Ann. Appl. Probab. 19 (2009), no. 1, 49--70. doi:10.1214/08-AAP531.

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