Indefinitely Oscillating Martingales

TL;DR

This paper constructs nonnegative martingales that oscillate indefinitely with high probability, providing bounds on their oscillations and exploring implications for hypothesis testing and description length limits.

Contribution

It introduces a class of martingales with high-probability indefinite oscillations and derives near-optimal bounds on their oscillation rates.

Findings

Oscillating martingales can oscillate indefinitely with high probability.

Derived bounds on the number of oscillations close to theoretical limits.

Applications include limits of description length operators and belief change frequency.

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.