Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The spread of a catalytic branching random walk

Philippe Carmona and Yueyun Hu

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Abstract

We consider a catalytic branching random walk on $\mathbb{Z} $ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_{n}$: For some constant $\alpha$, $\frac{M_{n}}{n}\to\alpha$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_{n}-\alpha n$ as $n$ goes to infinity.

Résumé

Nous considérons une marche aléatoire branchant catalytique sur $\mathbb{Z} $ qui ne branche qu’à l’origine. Dans le cas surcritique, nous établissons une loi des grands nombres pour la position maximale $M_{n}$ : Il existe une constante $\alpha$ explicite telle que $\frac{M_{n}}{n}\to\alpha$ presque sûrement sur l’ensemble des trajectoires pour lesquelles l’origine est visitée une infinité de fois.

Ensuite, nous déterminons toutes les lois limites possibles, lorsque $n\to+\infty$, pour la suite $M_{n}-\alpha n$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 327-351.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856130

Digital Object Identifier
doi:10.1214/12-AIHP529

Mathematical Reviews number (MathSciNet)
MR3189074

Zentralblatt MATH identifier
1291.60208

Subjects
Primary: 60K37: Processes in random environments

Keywords
Branching processes Catalytic branching random walk

Citation

Carmona, Philippe; Hu, Yueyun. The spread of a catalytic branching random walk. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 327--351. doi:10.1214/12-AIHP529. https://projecteuclid.org/euclid.aihp/1395856130


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