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By Peres Y., Zeitouni O.

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Springer, Heidelberg (1988) 13. : The method of averaging and walks in inhomogeneous environments. Russ. Math. Surv. 40, 73–145 (1985) 14. : Random walks and percolation on trees. 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.

2, 93–143 (2000) 24. : Long range estimates for Markov chains. Bull. Sci. Math. 109, 225–252 (1985) 25. : Random walks in random environment. XXXI Summer school in probability, St Flour (2001). Lecture notes in Mathematics, vol. 1837, pp. 193–312.

C ) ≤ b−c3 k . On the other hand, the event C Using (81), it follows that PGW (Ck,1 k,1 ∩ 1 1 c k/8 (Ck,1 ) implies that the sum of the difference τi+1 − τi , i = 1, . . , b , is larger 123 628 Y. Peres, O. Zeitouni than bk/4 /2, and hence, by Markov’s inequality, PGW (Ck,1 ∩ (Ck,1 )c ) ≤ 2bk/8 E GW (τ21 − τ11 ) ≤ b−c5 k , bk/4 for some deterministic constant c5 < c4 . Since the same estimates are valid also for Ck,2 and Ck,2 replacing Ck,1 and Ck,1 , it follows that PGW (Ckc ) ≤ 4b−c5 k . (82) On the other hand, let Z i denote the collection of vertices in D bk/8 hit by X ·i .

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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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