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On weighted branching processes in random environment

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  • Kuhlbusch, Dirk

Abstract

In this paper we investigate the nonnegative martingale Wn=Zn/[mu]n(U), n[greater-or-equal, slanted]0 and its a.s. limit W, when (Zn)n[greater-or-equal, slanted]0 is a weighted branching process in random environment with stationary ergodic environmental sequence U=(Un)n[greater-or-equal, slanted]0 and [mu]n(U) denotes the conditional expectation of Zn given U for n[greater-or-equal, slanted]0. We find necessary and sufficient conditions for W to be nondegenerate, generalizing earlier results in the literature on ordinary branching processes in random environment and also weighted branching processes. In the important special case of i.i.d. random environment, a Z log Z-condition turns out to be crucial. Deterministic and nonvarying environments are treated as special cases. Our arguments adapt the probabilistic proof of Biggins' theorem for branching random walks given by Lyons (1997) to our situation.

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  • Kuhlbusch, Dirk, 2004. "On weighted branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 113-144, January.
    • Handle: RePEc:eee:spapps:v:109:y:2004:i:1:p:113-144
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    References listed on IDEAS

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    1. Coffey, John & Tanny, David, 1984. "A necessary and sufficient condition for noncertain extinction of a branching process in a random environment (BPRE)," Stochastic Processes and their Applications, Elsevier, vol. 16(2), pages 189-197, February.
    2. Liu, Quansheng, 1999. "Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 61-87, July.
    3. Tanny, David, 1978. "Normalizing constants for branching processes in random environments (B.P.R.E.)," Stochastic Processes and their Applications, Elsevier, vol. 6(2), pages 201-211, January.
    4. Dekking, F. M., 1987. "On the survival probability of a branching process in a finite state i.i.d. environment," Stochastic Processes and their Applications, Elsevier, vol. 27, pages 151-157.
    5. Tanny, David, 1988. "A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means," Stochastic Processes and their Applications, Elsevier, vol. 28(1), pages 123-139, April.
    6. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
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    Cited by:

    1. Gao, Zhiqiang & Liu, Quansheng, 2016. "Exact convergence rates in central limit theorems for a branching random walk with a random environment in time," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2634-2664.
    2. Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    3. Huang, Chunmao & Liu, Quansheng, 2024. "Limit theorems for a branching random walk in a random or varying environment," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    4. Alsmeyer, Gerold & Gröttrup, Sören, 2016. "Branching within branching: A model for host–parasite co-evolution," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1839-1883.
    5. Xiaoqiang Wang & Chunmao Huang, 2017. "Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 961-995, September.
    6. Harris, S.C. & Roberts, M.I., 2009. "Measure changes with extinction," Statistics & Probability Letters, Elsevier, vol. 79(8), pages 1129-1133, April.

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