Beyond Independence: Limit Theorems for Martingales with Dependent Stopping Times

Many classical results (e.g., Doob's optional sampling theorem, central limit theorems) crucially rely on stopping times being independent or conditionally independent with respect to the filtration. However, as Protter & Quintos (2021) show, this assumption is often unnecessarily strong and not satisfied in real-world models like contagion or credit risk. Building on their construction of dependent stopping times, this research would characterize how such dependencies affect martingale convergence, maximal inequalities, and concentration bounds. Unlike Hadjikyriakou & Prakasa Rao (2025), who extend results to demimartingales but still hinge on monotonicity or weak dependence, this project would systematically analyze moderate and strong dependencies, potentially identifying new "correction terms" or modified conditions under which limit theorems hold. This could reveal subtle boundary behaviors and open up applications in areas where simultaneous or interacting stopping events are natural, such as systemic risk or network contagion.

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. Stopping times occurring simultaneously. P. Protter, Alejandra Quintos (2021). E S A I M: Probability & Statistics.
3. Dependent Stopping Times. P. Protter, Alejandra Quintos (2021). Social Science Research Network.