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By Peres Y., Zeitouni O.

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V∈D (83) bk/8 When λ > 1, there exists a constant c6 < c5 such that uniformly in v ∈ D bk/8 , PTo (X ·2 visits v before time bk/4 ) ≤ bk/4 e−c6 b . 2 of [10] shows that there exists a β > 0 such that with Mv = |{w is an ancestor of v : dw ≥ 2}|, it holds that lim sup PGW ( min Mv / < β) < 0. →∞ v∈D It immediately follows, reducing c6 if necessary, that when λ ≤ 1, for all k large, E GW max v∈D bk/8 PTo (X ·2 visits v ever) ≤ e−c6 b . k/8 Substituting in (83), we conclude that whenever λ < m, PGW (Ack ∩ Ck ) ≤ 2PGW ((A1k )c ∩ Ck ) ≤ e−c7 b .

Ann. Probab. 18, 931–958 (1990) 15. : Probability on trees and networks. iu. html 16. : Conceptual proofs of L log L iteria for mean behavior of branching processes. Ann. Probab. 23, 1125–1138 (1995) 17. : Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure. Ergod. Theory Dyn. Syst. 15, 593–619 (1995) 18. : Biased random walks on Galton–Watson trees. Probab. Theory Relat. Fields 106, 249–264 (1996) 19. : Harmonic moments and large deviation rates for superitical branching processes.

Probability on trees and networks. iu. html 16. : Conceptual proofs of L log L iteria for mean behavior of branching processes. Ann. Probab. 23, 1125–1138 (1995) 17. : Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure. Ergod. Theory Dyn. Syst. 15, 593–619 (1995) 18. : Biased random walks on Galton–Watson trees. Probab. Theory Relat. Fields 106, 249–264 (1996) 19. : Harmonic moments and large deviation rates for superitical branching processes. Ann. Appl.

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A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees by Peres Y., Zeitouni O.


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