Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

Xiaoqiang Wang and Chunmao Huang ()
Additional contact information
Xiaoqiang Wang: Shandong University (Weihai)
Chunmao Huang: Harbin Institute of Technology at Weihai

Journal of Theoretical Probability, 2017, vol. 30, issue 3, 961-995

Abstract: Abstract We consider a branching random walk on $${\mathbb {R}}$$ R with a stationary and ergodic environment $$\xi =(\xi _n)$$ ξ = ( ξ n ) indexed by time $$n\in {\mathbb {N}}$$ n ∈ N . Let $$Z_n$$ Z n be the counting measure of particles of generation n and $$\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)$$ Z ~ n ( t ) = ∫ e t x Z n ( d x ) be its Laplace transform. We show the $$L^p$$ L p convergence rate and the uniform convergence of the martingale $$\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]$$ Z ~ n ( t ) / E [ Z ~ n ( t ) | ξ ] , and establish a moderate deviation principle for the measures $$Z_n$$ Z n .

Keywords: Branching random walk; Random environment; Moment; $$L^p$$ L p convergence; Convergence rate; Uniform convergence; Moderate deviation; 60J80; 60K37; 60F10 (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10959-016-0668-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:30:y:2017:i:3:d:10.1007_s10959-016-0668-6

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-016-0668-6

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().