Adaptive Stopping in High Dimensions: Simulation-Driven Conjectures and Theory

As May et al. (2022) and Xie & Johnstone (2024) show, high-dimensional data analysis often pushes classical probabilistic tools to their limits, especially in the presence of complex dependencies and small samples. This research would develop and analyze simulation frameworks for high-dimensional martingale sequences and stopping times—potentially with structured dependence, as in Protter & Quintos (2021) or Bai-Silverstein-type martingale CLTs. By systematically varying dimensionality, dependence structure, and stopping criteria, the project would seek to identify patterns, anomalies, or phase transitions not yet captured theoretically. These observations could then motivate new conjectures about limit theorems, deviation inequalities, or optimal stopping boundaries in high dimensions, ultimately feeding back into new proofs and theory. This approach could significantly extend the practical utility of martingale methods in modern data science.

References:

1. Stopping times occurring simultaneously. P. Protter, Alejandra Quintos (2021). E S A I M: Probability & Statistics.
2. Combined Pruning for Nested Cross-Validation to Accelerate Automated Hyperparameter Optimization for Embedded Feature Selection in High-Dimensional Data with Very Small Sample Sizes. Sigrun May, Sven Hartmann, F. Klawonn (2022). arXiv.org.
3. CLT for Linear Spectral Statistics in High-Dimensional Random Effects Models. Ran Xie, Iain Johnstone (2024).