Integrated functionals of normal and fractional processes

Boris Buchmann, Ngai Hang Chan

Published 2009-02-27Version 1

Consider $Z^f_t(u)=\int_0^{tu}f(N_s) ds$, $t>0$, $u\in[0,1]$, where $N=(N_t)_{t\in\mathbb{R}}$ is a normal process and $f$ is a measurable real-valued function satisfying $Ef(N_0)^2<\infty$ and $Ef(N_0)=0$. 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 $t\to\infty$. 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 $t\to\infty$ 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 $H\in(0,1)$, 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 $H=3/4$.