Central Limit Theorems for Random Walks in Random Environments with Memory and Correlated Noises: Theory and Applications to Sequential Decision Making

We establish quenched and annealed functional central limit theorems for discrete-time random walks in random environments that exhibit temporal memory and spatially corre- lated noise. The environment is a stationary ergodic process satisfying á-mixing or path-cone mixing conditions. Under uniform ellipticity and moment assumptions, we prove that the scaled walk converges to a Brownian motion with an explicit diÐusion matrix. The proofs combine regeneration techniques adapted to dependent environments, martingale approxi- mation, and novel coupling arguments. We derive non-asymptotic Wasserstein bounds and apply the results to uncertainty quantiÑcation in algorithmic trading and reinforcement learning. A detailed case study on predicting unseen species using Good{Toulmin estima- tors illustrates the practical relevance. Numerical experiments with PyTorch conÑrm the theoretical predictions.