Rank-One Fluctuations in Averaging-Learning Dynamics

We study averaging-learning dynamics without an exogenous ground truth: the reference signal is generated endogenously by the population. The dynamics combine a time-varying averaging matrix, a learning-source matrix, and a learning matrix, typically diagonal. Dobrushin-type contraction controls the decay of oscillations and yields asymptotic agreement. Under a summability condition, the backward products converge exponentially to rank-one limits. With summable perturbations, the process converges to a random consensus state. For i.i.d. perturbations, we prove a central limit theorem for the centered process: the limiting Gaussian law is supported on the agreement direction. Thus, despite the multi-agent dynamics, the long-time fluctuations are asymptotically one-dimensional. We also record a pairwise Dobrushin formulation that clarifies the dynamic agreement-class geometry underlying the one-class regime.