The Annals of Applied Probability

Spreading speeds in reducible multitype branching random walk

J. D. Biggins

Full-text: Open access


This paper gives conditions for the rightmost particle in the $n$th generation of a multitype branching random walk to have a speed, in the sense that its location divided by $n$ converges to a constant as $n$ goes to infinity. Furthermore, a formula for the speed is obtained in terms of the reproduction laws. The case where the collection of types is irreducible was treated long ago. In addition, the asymptotic behavior of the number in the $n$th generation to the right of $na$ is obtained. The initial motive for considering the reducible case was results for a deterministic spatial population model with several types of individual discussed by Weinberger, Lewis and Li [J. Math. Biol. 55 (2007) 207–222]: the speed identified here for the branching random walk corresponds to an upper bound for the speed identified there for the deterministic model.

Article information

Ann. Appl. Probab., Volume 22, Number 5 (2012), 1778-1821.

First available in Project Euclid: 12 October 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes [See also 92Dxx] 92D25: Population dynamics (general)

Branching random walk multitype speed anomalous spreading reducible


Biggins, J. D. Spreading speeds in reducible multitype branching random walk. Ann. Appl. Probab. 22 (2012), no. 5, 1778--1821. doi:10.1214/11-AAP813.

Export citation


  • Asmussen, S. and Hering, H. (1983). Branching Processes. Progress in Probability and Statistics 3. Birkhäuser, Boston, MA.
  • Biggins, J. D. (1976a). The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 446–459.
  • Biggins, J. D. (1976b). Asymptotic properties of the branching random walk. Ph.D. Phil. thesis, Univ. Oxford.
  • Biggins, J. D. (1977). Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 630–636.
  • Biggins, J. D. (1995). The growth and spread of the general branching random walk. Ann. Appl. Probab. 5 1008–1024.
  • Biggins, J. D. (1997). How fast does a general branching random walk spread? In Classical and Modern Branching Processes (Minneapolis, MN, 1994). The IMA Volumes in Mathematics and Its Applications 84 19–39. Springer, New York.
  • Biggins, J. D. (2010). Branching out. In Probability and Mathematical Genetics. London Mathematical Society Lecture Note Series 378 113–134. Cambridge Univ. Press, Cambridge.
  • Biggins, J. D. and Rahimzadeh Sani, A. (2005). Convergence results on multitype, multivariate branching random walks. Adv. in Appl. Probab. 37 681–705.
  • Chen, L. H. Y. (1978). A short note on the conditional Borel–Cantelli lemma. Ann. Probab. 6 699–700.
  • Jagers, P. (1989). General branching processes as Markov fields. Stochastic Process. Appl. 32 183–212.
  • Kawata, T. (1972). Fourier Analysis in Probability Theory. Probability and Mathematical Statistics 15. Academic Press, New York.
  • Kingman, J. F. C. (1961). A convexity property of positive matrices. Quart. J. Math. Oxford Ser. (2) 12 283–284.
  • Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices, 2nd ed. Academic Press, Orlando, FL.
  • Miller, H. D. (1961). A convexivity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32 1260–1270.
  • Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
  • Rouault, A. (1987). Probabilités de présence dans un processus de branchement spatial markovien. Ann. Inst. Henri Poincaré Probab. Stat. 23 37–61.
  • Rouault, A. (1993). Precise estimates of presence probabilities in the branching random walk. Stochastic Process. Appl. 44 27–39.
  • Seneta, E. (1973). Non-Negative Matrices: An Introduction to Theory and Applications. Halsted, New York.
  • Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd ed. Springer, New York.
  • Weinberger, H. F., Lewis, M. A. and Li, B. (2007). Anomalous spreading speeds of cooperative recursion systems. J. Math. Biol. 55 207–222.