A remark on H1 martingales

The space of H1 martingales is interesting because of its duality with the space of BMO martingales. It is straightforward to show that every H1 martingale is a uniformly integrable martingale. However, the converse is not true. That is to say, some uniformly integrable martingales are not H1 martingales. This brief note provides a template for systematically constructing such processes.