Exact convergence rates in central limit theorems for a branching random walk with a random environment in time

Abstract

International audienceChen [Ann. Appl. Probab. 11 (2001), 1242–1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process. We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form $EX(\ln^+ X)^{ 1+λ} < ∞$. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work. The results are established in the more general framework, i.e. for a branching random walk with a random environment in time. The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment

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

“…In [18,Theorem 2.3], the authors proved the following result about the exact rate of convergence in the central limit theorem: if Em −δ 0 < ∞ for some δ > 0, (2.1) holds for some λ > 8 and (2.2) holds for some η > 12, then for all t ∈ R,…”

“…The model a branching random walk with a random environment in time can be formulated as follows [18,20]. Let (Θ, Ô) be a probability space, and (Θ N , Ô ⊗N ) = (Ω, τ ) be the corresponding product space.…”

“…Révész, Rosen and Shi (2005, [36]) found full asymptotic expansions in the local limit theorem for branching Wiener processes, while Grübel and Kabluchko (2015, [22]) obtained the similar result for a branching random walk on Z and discussed the related applications in random trees. The exact convergence rate obtained in [11,18] can be formulated as the first order asymptotic expansion in the central limit theorem for the models considered therein. Inspired by these works, we consider the following natural question: what about the asymptotic expansion of higher orders?The aim of this article is to derive the second and third orders asymptotic expansions in the central limit theorem for a branching random walk with a time-dependent random environment.…”

“…The obtained results and the developed methods can be used to obtain asymptotic expansions of orders 4, 5, etc., and hint the general formula for each finite order expansion, although we have not yet been able to prove it: see Conjecture 2.7 and the comments following it. We also mention that the approaches in our previous work [18] have been significantly developed in the present article.The article is organized as follows. In Section 2, after giving the rigorous definition of the model of a branching random walk with a random environment in time and introducing three martingales, we formulate the results on convergence rates of martingales as Theorem 2.1, and then state the main results on the asymptotic expansions in Theorems 2.3 and 2.4.…”

“…The obtained results and the developed methods can be used to obtain asymptotic expansions of orders 4, 5, etc., and hint the general formula for each finite order expansion, although we have not yet been able to prove it: see Conjecture 2.7 and the comments following it. We also mention that the approaches in our previous work [18] have been significantly developed in the present article.…”