Exact Convergence Rates for Particle Distributions in a Non-Lattice Branching Random Walk

Consider a discrete-time supercritical branching random walk, which is a branching process combined with random spatial motion of particles, the number of the descendants tending to infinity with positive probability. Let Z(n)(center dot) be the counting measure which counts the number of particles of generation n in a given set. Revesz (1994) studied the convergence rates in the central and local limit theorems for Z(n)(center dot) in some special cases, and then, the topic was further developed in various cases. In this paper, we give the exact convergence rates of the central and local limit theorems for Z(n) in the case the underlying motion is governed by a general non-lattice random walk on R-d with the characteristic function of the motion law satisfying the weak Cramer condition.