Project Euclid

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.
  Zentralblatt MATH: 0356.60053
  Digital Object Identifier: doi:10.1017/S0021900200104644
  
• [3] Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, (1992), 137–151.
  Zentralblatt MATH: 0748.60080
  Digital Object Identifier: doi:10.1214/aop/1176989921
  Project Euclid: euclid.aop/1176989921
  
• [4] Biggins, J.D. and Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, (2004), 544–581.
  Zentralblatt MATH: 1056.60082
  Digital Object Identifier: doi:10.1017/S0001867800013604
  Project Euclid: euclid.aap/1086957585
  
• [5] Bingham, N.H., Goldie, C.M. and Teugels, J.L.: Regular Variation, Cambridge Univ. Press, Cambridge, 1987.
  Zentralblatt MATH: 0617.26001
  
• [6] Chow, Y.S. and Teicher, H.: Probability theory: Independence, Interchangeability and Martingales, Springer-Verlag, New York, 1988.
  Zentralblatt MATH: 0652.60001
  
• [7] Durrett, R. and Liggett, T.: Fixed points of the smoothing transformation. Z. Wahrsch. verw. Geb. 64, (1983), 275–301.
  Zentralblatt MATH: 0506.60097
  Digital Object Identifier: doi:10.1007/BF00532962
  
• [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.
  Zentralblatt MATH: 1313.60023
  Digital Object Identifier: doi:10.1016/S0252-9602(14)60023-0
  
• [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.
  Zentralblatt MATH: 1343.60014
  Digital Object Identifier: doi:10.1016/j.spa.2016.02.013
  
• [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.
  Zentralblatt MATH: 1308.60098
  Digital Object Identifier: doi:10.1007/s11464-014-0407-1
  
• [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
  arXiv: 1903.00524
  Zentralblatt MATH: 07096910
  Digital Object Identifier: doi:10.1016/j.jmaa.2019.07.017
  
• [14] Kahane, J.P. and Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, (1976), 131–145.
  Zentralblatt MATH: 0349.60051
  Digital Object Identifier: doi:10.1016/0001-8708(76)90151-1
  
• [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.
  Zentralblatt MATH: 1359.60105
  Digital Object Identifier: doi:10.1214/17-ECP50
  
• [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.
  Zentralblatt MATH: 1288.60113
  Digital Object Identifier: doi:10.1134/S0081543813060126
  
• [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.
  Zentralblatt MATH: 1058.60068
  Digital Object Identifier: doi:10.1016/S0304-4149(01)00092-8
  
• [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.
  Zentralblatt MATH: 0897.60086
  
• [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.
  Zentralblatt MATH: 0499.60088
  Digital Object Identifier: doi:10.1214/aop/1176993712
  Project Euclid: euclid.aop/1176993712
  
• [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.
  Zentralblatt MATH: 1386.60298
  Digital Object Identifier: doi:10.1007/s10959-016-0668-6