Integrated functionals of normal and fractional processes

Consider Ztf(u)=.0tuf(Ns).ds, t>0, u.[0, 1], where N=(Nt)t.. 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.