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

详细信息    查看全文

• 作者：Zhiqiang Gao^a ; ^{gaozq@bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Quansheng Liu^b ; ^c ; ^{quansheng.liu@univ-ubs.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60K37 ; 60J10 ; 60F05 ; 60J80
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：126
• 期：9
• 页码：2634-2664
• 全文大小：415 K

文摘

Chen (2001) 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<&infin;iner hidden">$\mathbb{E}X{\left({ln}^{+}X\right)}^{1+\lambda }<\mathrm{\&<font\; color="red">in</font>f<font\; color="red">in</font>;}$. In the rate functions that we find for a branching random walk, we figure out some new terms which did not 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.