Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time (Q2405202)

The authors consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time. For the normalised counting measure of the number of particles of generation \(n\) in a given region, they give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher-order expansions and hint a general formula for each finite-order expansion, although they have not been able to prove it. In the proofs, a version of the Edgeworth expansion of central limit theorems for sums of independent random variables due to \textit{Z. Bai} and \textit{L. Zhao} [Sci. Sin., Ser. A 29, 1--22 (1986; Zbl 0629.60028)], truncating arguments and martingale approximation play key roles. In particular, they introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which may be of independent interest. As a special case of their results, they can derive the second and third order expansions for the branching Wiener process, where the underlying branching process is a Galton-Watson process whose offspring distribution has a mean greater than 1 and the motion of particles is governed by a Wiener process.