The Annals of Probability

The Seneta–Heyde scaling for the branching random walk

Elie Aidekon and Zhan Shi

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We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609–631] in a one-dimensional super-critical branching random walk, and study the additive martingale $(W_{n})$. We prove that, upon the system’s survival, $n^{1/2}W_{n}$ converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in Appl. Probab. 36 (2004) 544–581], of the derivative martingale.

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Ann. Probab., Volume 42, Number 3 (2014), 959-993.

First available in Project Euclid: 26 March 2014

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Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems

Branching random walk Seneta–Heyde norming additive martingale derivative martingale


Aidekon, Elie; Shi, Zhan. The Seneta–Heyde scaling for the branching random walk. Ann. Probab. 42 (2014), no. 3, 959--993. doi:10.1214/12-AOP809.

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  • [1] Addario-Berry, L. and Reed, B. (2009). Minima in branching random walks. Ann. Probab. 37 1044–1079.
  • [2] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362–1426.
  • [3] Aïdékon, E. and Jaffuel, B. (2011). Survival of branching random walks with absorption. Stochastic Process. Appl. 121 1901–1937.
  • [4] Aïdékon, E. and Shi, Z. (2010). Weak convergence for the minimal position in a branching random walk: A simple proof. Period. Math. Hungar. 61 43–54.
  • [5] Berestycki, J., Harris, S. C. and Kyprianou, A. E. (2011). Traveling waves and homogeneous fragmentation. Ann. Appl. Probab. 21 1749–1794.
  • [6] Bertoin, J. and Rouault, A. (2005). Discretization methods for homogeneous fragmentations. J. Lond. Math. Soc. (2) 72 91–109.
  • [7] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
  • [8] Biggins, J. D. (1998). Lindley-type equations in the branching random walk. Stochastic Process. Appl. 75 105–133.
  • [9] Biggins, J. D. (2010). Branching out. In Probability and Mathematical Genetics (N. H. Bingham and C. M. Goldie, eds.). London Mathematical Society Lecture Note Series 378 113–134. Cambridge Univ. Press, Cambridge.
  • [10] Biggins, J. D. and Kyprianou, A. E. (1996). Branching random walk: Seneta–Heyde norming. In Trees (Versailles, 1995) (B. Chauvin et al., eds.). Progress in Probability 40 31–49. Birkhäuser, Basel.
  • [11] Biggins, J. D. and Kyprianou, A. E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Probab. 25 337–360.
  • [12] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544–581.
  • [13] Biggins, J. D. and Kyprianou, A. E. (2005). Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 609–631.
  • [14] Borovkov, A. A. and Foss, S. G. (2000). Estimates for the excess of a random walk over an arbitrary boundary and their applications. Theory Probab. Appl. 44 231–253.
  • [15] Chauvin, B. and Rouault, A. (1988). KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Related Fields 80 299–314.
  • [16] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 2nd ed. Wiley, New York.
  • [17] Harris, S. C. (1999). Travelling-waves for the FKPP equation via probabilistic arguments. Proc. Roy. Soc. Edinburgh Sect. A 129 503–517.
  • [18] Heyde, C. C. (1970). Extension of a result of Seneta for the super-critical Galton–Watson process. Ann. Math. Statist. 41 739–742.
  • [19] Hu, Y. and Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 742–789.
  • [20] Jaffuel, B. (2012). The critical barrier for the survival of the branching random walk with absorption. Ann. Inst. H. Poincaré Probab. Statist. 48 989–1009.
  • [21] Kahane, J. P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 131–145.
  • [22] Kozlov, M. V. (1976). The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment. Theory Probab. Appl. 21 791–804.
  • [23] Kyprianou, A. E. (2004). Travelling wave solutions to the K–P–P equation: Alternatives to Simon Harris’ probabilistic analysis. Ann. Inst. Henri Poincaré Probab. Stat. 40 53–72.
  • [24] Lai, T. L. (1976). Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Probab. 4 51–66.
  • [25] Lalley, S. P. and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 1052–1061.
  • [26] Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab. 30 85–112.
  • [27] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) (K. B. Athreya and P. Jagers, eds.). IMA Vol. Math. Appl. 84 217–221. Springer, New York.
  • [28] Lyons, R. and Peres, Y. (2010). Probability on Trees and Networks. Cambridge Univ. Press. Preprint. Available at
  • [29] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • [30] McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28 323–331.
  • [31] McKean, H. P. (1976). A correction to: “Application of Brownian motion to the equation of Kolmogorov–Petrovskiĭ–Piskonov” (Comm. Pure Appl. Math. 28 (1975) 323–331). Comm. Pure Appl. Math. 29 553–554.
  • [32] Mogul’skiĭ, A. A. (1973). Absolute estimates for moments of certain boundary functionals. Theory Probab. Appl. 18 340–347.
  • [33] Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. Henri Poincaré Probab. Stat. 22 199–207.
  • [34] Seneta, E. (1968). On recent theorems concerning the supercritical Galton–Watson process. Ann. Math. Statist. 39 2098–2102.