  In this article we prove large deviations principles for high minima of
Gaussian processes with nonnegatively correlated increments on arbitrary
intervals. Furthermore, we prove large deviations principles for the increments
of such processes on intervals $[a,b]$ where $b-a$ is either less than the
increment or twice the increment, assuming stationarity of the increments. As a
chief example, we consider fractional Brownian motion and fractional Gaussian
noise for $H\geq 1/2$.
