Integrated functionals of normal and fractional processes

Abstract

Consider Z_t^f(u)=∫₀^tuf(N_s) ds, t>0, u∈[0, 1], where N=(N_t)_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 Z_t^f/t^1/2 converges in distribution to a multiple of standard Brownian motion as t→∞. If the dependence is sufficiently strong, then Z_t/(EZ_t(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.