In my research group, we’re deeply engaged in studying limit theorems in probability, which play a crucial role in simplifying and understanding complex stochastic systems. Limit theorems are powerful because they allow us to focus on the essential spatial and temporal scales of these systems, stripping away much of the noise and intricacies to reveal the “first-order” behaviors that really drive the dynamics. By capturing these dominant features, we can gain clearer insights into how stochastic systems evolve over time.
One area where we’ve made significant contributions is in developing some of the earliest corrected diffusion approximations. These approximations go beyond traditional methods, providing more accurate models that account for higher-order effects. This is particularly useful in systems where small-scale variations can have a large impact on overall behavior.
These types of methods also arise in the study of small sample asymptotics for the distribution of rewards in reinforcement learning in control with small discount rates (note that emphasis on distributions and not only expected values!) These kinds of results are crucial for problems in areas like economics and operations research, where decisions often need to be made based on limited data and our group has developed key results to understand these stochastic objects.
The beauty of limit theorems is that they provide a bridge between complex stochastic models and more tractable, insightful approximations, allowing us to understand these systems more deeply while keeping the analysis manageable. We’re continuing to explore new applications of these ideas across a wide range of domains.