Electronic Journal of Probability

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

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

Rights
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