## Electronic Communications in Probability

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

#### 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
Accepted: 9 June 2019
First available in Project Euclid: 3 July 2019

https://projecteuclid.org/euclid.ecp/1562119369

Digital Object Identifier
doi:10.1214/19-ECP247

#### 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

#### 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.