Desvios moderados para o passeio aleatório minimal

Abstract

This dissertation investigates the asymptotic behavior of the minimal random walk (MRW), a non-Markovian stochastic model characterized by long-range memory. The dynamics of the process follow an evolution rule in which, at each step, a past time instant is chosen uniformly at random, and the walker replicates the action taken at that moment (either moving or remaining stationary). We demonstrate how the memory parameter α governs the macroscopic behavior of the process, determining the influence of the past trajectory on the probability of future movements. To analyze such behaviors, we establish a formal connection between the MRW and Pólya urn schemes, enabling the application of the spectral theory of substitution matrices to prove limit theorems. We show that for α < 1/2, the process lies in the diffusive regime, converging to a Gaussian process with dependent increments. At the critical point α = 1/2, the memory becomes asymptotically irrelevant, and the process converges to standard Brownian motion. In the superdiffusive regime (α > 1/2), the memory dominates the evolution, and convergence occurs toward a unique non-Gaussian random variable. Additionally, we analyze the “laziest” walk case (q = 0), in which the classical strong law of large numbers fails, and the process converges to a random limit with a Mittag-Leffler distribution. The main original contribution of this work is the proof of a moderate deviations principle (MDP) for the MRW in the diffusive regime (α < 1/2). Employing martingale theory, we determine the rate function that characterizes the probability of such deviations and their respective rate of convergence. This result fills a fundamental gap between typical fluctuations, described by the central limit theorem, and extreme events, described by large deviations theory, providing a rigorous and complete characterization of the model’s fluctuations.