PATH REGULARITY OF GAUSSIAN PROCESSES VIA SMALL
DEVIATIONS
Abstract: We study the a.s. sample path regularity of Gaussian processes. To this end we relate
the path regularity directly to the theory of small deviations. In particular, we show that
if the process is -times differentiable, then the exponential rate of decay of its
small deviations is at most . We also show a similar result if is not an
integer.
Further generalizations are given, which parallel the entropy method to determine the
small deviations. In particular, the present approach seems to be a probabilistic interpretation
of the multiplicativity property of the entropy numbers.
2000 AMS Mathematics Subject Classification: Primary: 60F99; Secondary:
60G15.
Keywords and phrases: Small deviation, small ball probability, Gaussian process,
sample path regularity.