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 $$ Z_n(\cdot ) $$ 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 $$ Z_n(\cdot ) $$ 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 $$ Z_n(\cdot ) $$ in the case the underlying motion is governed by a general non-lattice random walk on $${\mathbb {R}}^d$$ with the characteristic function of the motion law satisfying the weak Cramer condition.