  We consider the persistence probability of a certain fractional Gaussian
process $M^H$ that appears in the Mandelbrot-van Ness representation of
fractional Brownian motion. This process is self-similar and smooth. We show
that the persistence exponent of $M^H$ exists and is continuous in the Hurst
parameter $H$. Further, the asymptotic behaviour of the persistence exponent
for $H\downarrow0$ and $H\uparrow1$, respectively, is studied. Finally, for
$H\to 1/2$, the suitably renormalized process converges to a non-trivial limit
with non-vanishing persistence exponent, contrary to the fact that $M^{1/2}$
vanishes.
