Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 24 (2019), paper no. 41, 14 pp.
Convergence of complex martingale for a branching random walk in a time random environment
Xiaoqiang Wang and Chunmao Huang
Abstract
We consider a discrete-time branching random walk in a stationary and ergodic environment $\xi =(\xi _{n})$ indexed by time $n\in \mathbb{N} $. Let $W_{n}(z)$ ($z\in \mathbb{C} ^{d}$) be the natural complex martingale of the process. We show sufficient conditions for its almost sure and quenched $L^{\alpha }$ convergence, as well as the existence of quenched moments and weighted moments of its limit, and also describe the exponential convergence rate.
Article information
Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 41, 14 pp.
Dates
Received: 6 February 2019
Accepted: 9 June 2019
First available in Project Euclid: 3 July 2019
Permanent link to this document
https://projecteuclid.org/euclid.ecp/1562119369
Digital Object Identifier
doi:10.1214/19-ECP247
Mathematical Reviews number (MathSciNet)
MR3978690
Zentralblatt MATH identifier
07088982
Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments
Keywords
branching random walk random environment complex martingale moments weighted moments convergence rate
Rights
Creative Commons Attribution 4.0 International License.
Citation
Wang, Xiaoqiang; Huang, Chunmao. Convergence of complex martingale for a branching random walk in a time random environment. Electron. Commun. Probab. 24 (2019), paper no. 41, 14 pp. doi:10.1214/19-ECP247. https://projecteuclid.org/euclid.ecp/1562119369