Integrated functionals of normal and fractional processes

Consider Z^f_t(u) = ∫^tu₀ f(N_s)ds, t > 0, u ∈ [0, 1], where N = (N_t) _tℝ is a normal process and f is a measurable real-valued function satisfying Ef(N₀)² < ∞ and Ef(N ₀) = 0. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191-216] showed that Z^f_t/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.