Electronic Journal of Probability

Upper large deviations for Branching Processes in Random Environment with heavy tails

Vincent Bansaye and Christian Böinghoff

Full-text: Open access

Abstract

Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where 'in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n \gt 0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of Böinghoff & Kersting (2009) and Bansaye & Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 69, 1900-1933.

Dates
Accepted: 19 October 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820238

Digital Object Identifier
doi:10.1214/EJP.v16-933

Mathematical Reviews number (MathSciNet)
MR2851050

Zentralblatt MATH identifier
1245.60081

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K37: Processes in random environments 60J05: Discrete-time Markov processes on general state spaces 92D25: Population dynamics (general)

Keywords
Branching processes random environment large deviations random walks heavy tails

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bansaye, Vincent; Böinghoff, Christian. Upper large deviations for Branching Processes in Random Environment with heavy tails. Electron. J. Probab. 16 (2011), paper no. 69, 1900--1933. doi:10.1214/EJP.v16-933. https://projecteuclid.org/euclid.ejp/1464820238


Export citation

References

  • V.I. Afanasyev. Limit theorems for a conditional random walk and some applications. MSU. Diss. Cand. Sci. Moscow, 1980
  • V.I. Afanasyev, J. Geiger, G. Kersting and V.A. Vatutin. Functional limit theorems for strongly subcritical branching processes in random environment. Stochastic Process. Appl. 115 (2005), 1658-1676.
  • K.B. Athreya and S. Karlin. On branching processes with random environments: I, II. Ann. Math. Stat. 42 (1971), 1499-1520, 1843-1858.
  • V.I. Afanasyev, J. Geiger, G. Kersting and V.A. Vatutin. Criticality for branching processes in random environment. Ann. Probab. 33 (2005), 645-673.
  • V. Bansaye and J. Berestycki. Large deviations for branching processes in random environment. Markov Process. Related Fields 15 (2009), 493-524.
  • J. D. Biggins and N. H. Bingham. Large deviations in the supercritical branching process. Adv. in Appl. Probab. 25 (1993), 757-772.
  • N. H. Bingham and C. M. Goldie, J. L. Teugels. Regular Variation. Cambridge University Press (1987) Cambridge.
  • M. Birkner, J. Geiger and G. Kersting. Branching processes in random environment - a view on critical and subcritical cases. Interacting stochastic systems, Springer (2005) Berlin, 265-291.
  • C. Böinghoff and G. Kersting. Upper large deviations of branching in a random environment - Offspring distributions with geometrically bounded tails. Stoch. Proc. Appl. 120 (2010), 2064-2077.
  • A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Jones and Barlett Publishers International (1993) London.
  • J. Geiger and G. Kersting. The survival probability of a critical branching process in random environment. Theory Probab. Appl. 45 (2000), 517-525.
  • J. Geiger, G. Kersting and V.A. Vatutin. Limit theorems for subcritical branching processes in random environment. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), 593-620.
  • W. Feller. An Introduction to Probability Theory and Its Applications- Volume II. John Wiley & Sons, Inc. (1966) New York.
  • Y. Guivarc'h, Q. Liu. Asymptotic properties of branching processes in random environment. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 339-344.
  • F. den Hollander. Large Deviations. Fields Institute Monographs (2000) Providence, RI.
  • O. Kallenberg. Foundations of Modern Probability. Springer (2001) London.
  • M. V. Kozlov. On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment. Theory Probab. Appl. 21 (1976), 791-804.
  • M. V. Kozlov. On large deviations of branching processes in a random environment: geometric distribution of descendants. Discrete Math. Appl. 16 (2006), 155-174.
  • M. V. Kozlov. On Large Deviations of Strictly Subcritical Branching Processes in a Random Environment with Geometric Distribution of Progeny. Theory Probab. Appl. 54 (2010), 424-446.
  • A. Rouault. Large deviations and branching processes. Pliska Stud. Math. Bulgar. 13 (2000), 15-38.
  • W. L. Smith and W.E. Wilkinson. On branching processes in random environments. Ann. Math. Stat. 40 (1969), 814-824.