The spread of a catalytic branching random walk

Carmona, Philippe; Hu, Yueyun

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 327-351 / Harvested from Numdam

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Abstract

Nous considérons une marche aléatoire branchant catalytique sur $ℤ$ 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 $α$ explicite telle que $\frac{M_{n}}{n} \to α$ 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 $M_{n} - α n$ .

We consider a catalytic branching random walk on $ℤ$ 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 $α$ , $\frac{M_{n}}{n} \to α$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_{n} - α n$ as $n$ goes to infinity.

Publié le : 2014-01-01

MR 3189074 | Zbl 1291.60208

DOI : https://doi.org/10.1214/12-AIHP529
Classification: 60K37

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     author = {Carmona, Philippe and Hu, Yueyun},
     title = {The spread of a catalytic branching random walk},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
     pages = {327-351},
     doi = {10.1214/12-AIHP529},
     mrnumber = {3189074},
     zbl = {1291.60208},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2014__50_2_327_0}
}

Carmona, Philippe; Hu, Yueyun. The spread of a catalytic branching random walk. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 327-351. doi : 10.1214/12-AIHP529. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_2_327_0/

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