Martingale marginals do not always determine convergence

TL;DR

This paper demonstrates that martingale marginals alone do not guarantee convergence, providing explicit examples of martingales with identical marginals but different convergence behaviors.

Contribution

It constructs martingales with bounded increments that show convergence in distribution without almost sure convergence, highlighting limitations of marginals in determining convergence.

Findings

Constructed a martingale with bounded increments that converges in distribution but not almost surely.

Provided two martingales with identical marginals, one converging almost surely and the other not converging in probability.

Showed that marginals alone do not determine the convergence type of martingales.

Abstract

Baez-Duarte (1971) and Gilat (1972) gave examples of martingales that converge in probability (and hence in distribution) but not almost surely. Here such a martingale is constructed with uniformly bounded increments, and a construction is provided of two martingales with the same marginals, one of which converges almost surely, while the other does not converge in probability.