  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
$H=3/4$.
