Indefinitely Oscillating Martingales
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 DOI:
 10.48550/arXiv.1408.3169
 arXiv:
 arXiv:1408.3169
 Bibcode:
 2014arXiv1408.3169L
 Keywords:

 Computer Science  Machine Learning;
 Mathematics  Probability;
 Mathematics  Statistics Theory
 EPrint:
 ALT 2014, extended technical report