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Consider a branching random walk in which the offspring distribution and themoving law both depend on an independent and identically distributed randomenvironment indexed by the time.For the normalised counting measure of thenumber of particles of generation $n$ in a given region, we give the second andthird orders asymptotic expansions of the central limit theorem under ratherweak assumptions on the moments of the underlying branching and moving laws.The obtained results and the developed approaches shed light on higher orderexpansions. In the proofs, the Edgeworth expansion of central limit theoremsfor sums of independent random variables, truncating arguments and martingaleapproximation play key roles. In particular, we introduce a new martingale,show its rate of convergence, as well as the rates of convergence of some knownmartingales, which are of independent interest.

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