  We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not
exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$.
The probability $p(T)$ of such event is frequently a regularly varying function
at $\infty$ with exponent $\theta$. In applications this parameter can provide
information on fractal properties of processes that are subordinate to
$x(\cdot)$. For this reason the estimation of $\theta$ is an important
theoretical problem. Here, we consider the process $x(t)$ whose derivative is
fractional Brownian motion with self-similarity parameter $0<H<1$. For this
case we produce new computational evidence in favor of the relations $\log
p(T)=-\theta \log T(1+o(1))$ and $\theta =H(1-H)$. The estimates of $\theta$
are to within 0.01 in the range $0.1\le H\le 0.9$. An analytical result for the
problem in hand is known for the markovian case alone, i.e., for $H=1/2$. We
point out other statistics of $x(t)$ whose small values have probabilities of
the same order as $p(T)$ in the $\log$ scale.
