Bernoulli

On the survival probability for a class of subcritical branching processes in random environment

Vincent Bansaye and Vladimir Vatutin

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Abstract

Let $Z_{n}$ be the number of individuals in a subcritical Branching Process in Random Environment (BPRE) evolving in the environment generated by i.i.d. probability distributions. Let $X$ be the logarithm of the expected offspring size per individual given the environment. Assuming that the density of $X$ has the form

\[p_{X}(x)=x^{-\beta-1}l_{0}(x)e^{-\rho x}\] for some $\beta>2$, a slowly varying function $l_{0}(x)$ and $\rho\in(0,1)$, we find the asymptotic of the survival probability $\mathbb{P}(Z_{n}>0)$ as $n\rightarrow\infty$, prove a Yaglom type conditional limit theorem for the process and describe the conditioned environment. The survival probability decreases exponentially with an additional polynomial term related to the tail of $X$. The proof uses in particular a fine study of a random walk (with negative drift and heavy tails) conditioned to stay positive until time $n$ and to have a small positive value at time $n$, with $n\rightarrow\infty$.

Article information

Source
Bernoulli Volume 23, Number 1 (2017), 58-88.

Dates
Received: December 2013
Revised: March 2015
First available in Project Euclid: 27 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1475001348

Digital Object Identifier
doi:10.3150/15-BEJ723

Mathematical Reviews number (MathSciNet)
MR3556766

Zentralblatt MATH identifier
06673471

Keywords
branching processes heavy tails random environment random walks speed of extinction

Citation

Bansaye, Vincent; Vatutin, Vladimir. On the survival probability for a class of subcritical branching processes in random environment. Bernoulli 23 (2017), no. 1, 58--88. doi:10.3150/15-BEJ723. https://projecteuclid.org/euclid.bj/1475001348.


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References

  • [1] Afanasyev, V.I. (1998). Limit theorems for a moderately subcritical branching process in a random environment. Diskret. Mat. 10 141–157.
  • [2] Afanasyev, V.I., Böinghoff, C., Kersting, G. and Vatutin, V.A. (2012). Limit theorems for weakly subcritical branching processes in random environment. J. Theoret. Probab. 25 703–732.
  • [3] Afanasyev, V.I., Böinghoff, C., Kersting, G. and Vatutin, V.A. (2014). Conditional limit theorems for intermediately subcritical branching processes in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 50 602–627.
  • [4] Afanasyev, V.I., Geiger, J., Kersting, G. and Vatutin, V.A. (2005). Criticality for branching processes in random environment. Ann. Probab. 33 645–673.
  • [5] Afanasyev, V.I., Geiger, J., Kersting, G. and Vatutin, V.A. (2005). Functional limit theorems for strongly subcritical branching processes in random environment. Stochastic Process. Appl. 115 1658–1676.
  • [6] Athreya, K.B. and Karlin, S. (1971). On branching processes with random environments: I, II. Ann. Math. Stat. 42 1499–1520, 1843–1858.
  • [7] Bansaye, V. and Vatutin, V. (2014). Random walk with heavy tail and negative drift conditioned by its minimum and final values. Markov Process. Related Fields 20 633–652.
  • [8] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [9] Birkner, M., Geiger, J. and Kersting, G. (2005). Branching processes in random environment – A view on critical and subcritical cases. In Interacting Stochastic Systems 269–291. Berlin: Springer.
  • [10] Geiger, J., Kersting, G. and Vatutin, V.A. (2003). Limit theorems for subcritical branching processes in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 39 593–620.
  • [11] Guivarc’h, Y. and Liu, Q. (2001). Propriétés asymptotiques des processus de branchement en environnement aléatoire. C. R. Acad. Sci. Paris Sér. I Math. 332 339–344.
  • [12] Petrov, V.V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. New York: Clarendon Press.
  • [13] Smith, W.L. and Wilkinson, W.E. (1969). On branching processes in random environments. Ann. Math. Statist. 40 814–827.
  • [14] Vatutin, V. and Zheng, X. (2012). Subcritical branching processes in a random environment without the Cramer condition. Stochastic Process. Appl. 122 2594–2609.