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Consistent Minimal Displacement of Branching Random Walks
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Ming Fang, University of Minnesota
Ofer Zeitouni, University of Minnesota and Weizmann Institute
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Abstract
Let T denote a rooted b-ary tree and let {Sv}v∈ 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 Λ(.) Let mn denote the minimum of the variables Sv over all vertices at the nth generation, denoted by Dn. Under mild conditions, mn/n converges almost surely to a constant, which for convenience may be taken to be 0. With $bar Sv=max{S_w: w is on the geodesic connecting the root
to v}, define Ln=minv∈ Dn bar Sv. We prove that Ln/n1/3 converges almost surely to an explicit constant l0. This answers a question of Hu and Shi.
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Full text: PDF
Pages: 106-118
Published on: March 29, 2010
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Bibliography
- Addario-Berry, Louigi; Reed, Bruce. Minima in branching random walks. Ann. Probab. 37 (2009), no. 3, 1044--1079. MR2537549
- Bramson, Maury D. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978), no. 5, 531--581. MR0494541 (58 #13382)
- Bramson, Maury D. Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 (1978), no. 2, 89--108. MR0510529 (80d:60105)
- Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications.Second edition.Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2 MR1619036 (99d:60030)
- Hu, Yueyun; Shi, Zhan. Slow movement of random walk in random environment on a regular tree. Ann. Probab. 35 (2007), no. 5, 1978--1997. MR2349581 (2009d:60340)
- G. Faraud, Y. Hu and Z. Shi, An almost sure convergence for stochastically biased random walks on trees, personal communication (2009).
- Mogulʹskiĭ, A. A. Small deviations in the space of trajectories.(Russian) Teor. Verojatnost. i Primenen. 19 (1974), 755--765. MR0370701 (51 #6927)
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Electronic Communications in Probability. ISSN: 1083-589X |
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