  We construct a class of nonnegative martingale processes that oscillate
indefinitely with high probability. For these processes, we state a uniform
rate of the number of oscillations and show that this rate is asymptotically
close to the theoretical upper bound. These bounds on probability and
expectation of the number of upcrossings are compared to classical bounds from
the martingale literature. We discuss two applications. First, our results
imply that the limit of the minimum description length operator may not exist.
Second, we give bounds on how often one can change one's belief in a given
hypothesis when observing a stream of data.
