Electronic Communications in Probability

Convergence of complex martingale for a branching random walk in a time random environment

Xiaoqiang Wang and Chunmao Huang

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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

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


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References

  • [1] Alsmeyer, G., Iksanov, A., Polotskiy, S. and Rösler, U.: Exponential rate of $L_{p}$-convergence of instrinsic martingales in supercritical branching random walks. Theory Stoch. Process 15, (2009), 1–18.
  • [2] Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. prob. 14, (1977), 25–37.
  • [3] Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, (1992), 137–151.
  • [4] Biggins, J.D. and Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, (2004), 544–581.
  • [5] Bingham, N.H., Goldie, C.M. and Teugels, J.L.: Regular Variation, Cambridge Univ. Press, Cambridge, 1987.
  • [6] Chow, Y.S. and Teicher, H.: Probability theory: Independence, Interchangeability and Martingales, Springer-Verlag, New York, 1988.
  • [7] Durrett, R. and Liggett, T.: Fixed points of the smoothing transformation. Z. Wahrsch. verw. Geb. 64, (1983), 275–301.
  • [8] Gao, Z., Liu, Q. and Wang, H.: Central limit theorems for a branching random walk with a random environment in time. Acta Math. Sci. 34, B (2) (2014), 501–512.
  • [9] Gao, Z. and Liu, Q.: Exact convergence rates in central limit theorems for a branching random walk with a random environment in time. Stoch. Proc. Appl. 126, (2016), 2634–2664.
  • [10] Guivarc’h, Y.: Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré. Probab. Statist. 26, (1990), 261–285.
  • [11] Huang, C., Liang, X. and Liu, Q.: Branching random walks with random environments in time, Front. Math. China 9, (2014), 835–842.
  • [12] Huang, C., and Liu, Q.: Convergence in $L^{p}$ and its exponential rate for a branching process in a random environment. Electro. J. Probab. 19, (2014), no. 104, 1–22.
  • [13] Iksanov, A., Liang, X. and Liu, Q.: On $L^{p}$-convergence of the Biggins martingale with complex parameter, arXiv:1903.00524
  • [14] Kahane, J.P. and Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, (1976), 131–145.
  • [15] Kolesko, K. and Meiners, M.: Convergence of complex martingales in the branching random walk: the boundary. Electron. Commun. Probab. 22, (2017), no. 18, 1–14.
  • [16] Liang, X. and Liu, Q.: Weighted moments of the limit of a branching process in a random environment. Proceedings of the Steklov Institute of Mathematics 282, (2013), 127–145.
  • [17] Liang, X. and Liu, Q.: Weighted moments for Mandelbrot’s martingales. Electron. Commun. Probab. 20, (2015), no. 85, 1–12.
  • [18] Liu, Q.: On generalized multiplicative cascades. Stoch. Proc. Appl. 86, (2000), 61–87.
  • [19] Liu, Q.: Asymptotic properties absolute continuity of laws stable by random weighted mean. Stoch. Proc. Appl. 95, (2001), 83–107.
  • [20] Lyons, R.: A simple path to Biggins’s martingale convergence for branching random walk. In K.B. Athreya, P. Jagers, (eds.), Classical and Modern Branching Processes, IMA Vol. Math. Appl. 84, 217–221, Springer-Verlag, New York, 1997.
  • [21] Mallein, B. and Miloś, P.: Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment. Stoch. Proc. Appl., doi: 10.1016/j.spa.2018.09.008.
  • [22] Uchiyama, K.: Spatial growth of a branching process of particles living in $\mathbb{R} ^{d}$, Ann. Probab. 10, (1982), no.4, 896–918.
  • [23] Wang, X. and Huang, C.: Convergence of martingale and moderate deviations for a branching random walk with a random environment in time. J. Theor. Probab. 30, (2017), 961–995.