Scaling limit in local time of near-critical nearest random walk

In 1961, Lamperti proves that a sequence of certain near-critical nearest random walks converges weakly to Brownian motion after proper scaling. Scaling limit in local times is then considered. We prove that local times converge to those of Brownian motion by corresponding scaling. Our proof is based on intrinsic branching structure of random walk and convergence of time in homogeneous branching processe.