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Consistent Minimal Displacement of Branching Random Walks


 
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1. Title Title of document Consistent Minimal Displacement of Branching Random Walks
 
2. Creator Author's name, affiliation, country Ming Fang; University of Minnesota
 
2. Creator Author's name, affiliation, country Ofer Zeitouni; University of Minnesota and Weizmann Institute
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Branching Random Walk; Consistent Minimal Displacement
 
3. Subject Subject classification 60G50; 60J80
 
4. Description Abstract Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\Lambda(\cdot)$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by $\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be $0$. With $\bar S_v=\max\{S_w: w$ is on the geodesic connecting the root to $v \}$, define $L_n=\min_{v\in \mathbb{D}_n} \bar S_v$. We prove that $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSF; Weizmann
 
7. Date (YYYY-MM-DD) 2010-03-29
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1533
 
10. Identifier Digital Object Identifier 10.1214/ECP.v15-1533
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 15
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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