On the asymptotic analysis of lazy reinforced random walks: A martingale approach

We provide a comprehensive characterization of the limiting behavior of lazy reinforced random walks (LRRW's). These random walks exhibit three distinct phases: diffusive, critical, and superdiffusive. Using a martingale theory approach, we establish proper versions of the law of large numbers, the almost sure convergence to even moments of Gaussian distribution, the law of the iterated logarithm, the almost sure central limit theorem, and the functional central limit theorem for the diffusive and critical regimes. In the superdiffusive regime, we demonstrate strong convergence to a random variable, as well as a central limit theorem and a law of the iterated logarithm for the fluctuations.