Martingale Deviation Theory under Minimal Integrability: When Classical Assumptions Break Down

Classical martingale inequalities (Azuma-Hoeffding, Doob’s maximal) and limit theorems require integrability—usually square-integrable martingales. But what happens if we challenge this core assumption? Inspired by the suggestion in the "Challenge core assumptions" heuristic and the work of Elmansouri & Otmani (2025), who handle GRBSDEs with only $L^2$ data, this idea pushes further: systematically explore deviation and convergence results for martingales under only $L^1$, or even weaker, integrability. Are there "weak" analogues of concentration inequalities? How does the failure of integrability manifest in stopping time theorems? This could lead to a better understanding of processes with heavy-tailed increments or "explosive" boundary behaviors, complementing the demimartingale framework of Hadjikyriakou & Prakasa Rao (2025). It might also motivate new robust statistical methods for heavy-tailed data, where classical martingale tools break down.

References:

1. Doob's type optional sampling theorems and a central limit theorem for demimartingales with applications to associated sequences. M. Hadjikyriakou, B.L.S Prakasa Rao (2025).
2. Generalized Reflected BSDEs with RCLL Random Obstacles in a General Filtration. Badr Elmansouri, M. Otmani (2025).