Universal and complete sets in martingale theory

TL;DR

This paper characterizes the sets of divergence of martingales, showing they can represent any measure-zero G_delta_sigma set, and explores the complexity of convergence sets within the projective hierarchy.

Contribution

It establishes the universality and completeness of divergence sets in martingale theory, linking them to descriptive set theory and the projective hierarchy.

Findings

Any measure-zero G_delta_sigma set can be realized as a divergence set of a martingale.

The set of everywhere converging martingales is ${\bf\Pi}^1_1$-complete.

Provides universal and complete sets for classes ${\bf\Pi}^1_1$ and ${\bf\Sigma}^1_2$ in martingale theory.

Abstract

The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\_{\delta\sigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale. In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is ${\bf\Pi}^1\_1$-complete, in a uniform way. We derive from this some universal and complete sets for the whole projective hierarchy, via a general method. We provide some other complete sets for the classes ${\bf\Pi}^1\_1$ and ${\bf\Sigma}^1\_2$ in the theory of martingales.