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

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Upper large deviations for Branching Processes in Random Environment with heavy tails
2.	Creator	Author's name, affiliation, country	Vincent Bansaye; Ecole Polytechnique, Palaiseau
2.	Creator	Author's name, affiliation, country	Christian Böinghoff; Goethe-University Frankfurt/Main
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Branching processes, random environment, large deviations, random walks, heavy tails
3.	Subject	Subject classification	60J80; 60K37; 60J05; 92D25
4.	Description	Abstract	Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where \lq 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 $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n>0)/n$ 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 B\"oinghoff $\&$ 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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	German Research Foundation (DFG) and ANR Manege
7.	Date	(YYYY-MM-DD)	2011-10-19
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/933
10.	Identifier	Digital Object Identifier	10.1214/EJP.v16-933
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 16
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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