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Ann. Inst. H. Poincare 39, 527–555 (2003) 9. : A transmutation formula for Markov chains. Bull Sci. Math. 109, 399–405 (1985) 10. : Large deviations for random walks on Galton–Watson trees: averaging and uncertainty. Probab. Theory Relat. Fields 122, 241–288 (2001) 11. : Random walks and electric networks. Carus Mathematical Monographs, vol. 22. Mathematical Association of America, Washington, DC (1984) 12. : Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Heidelberg (1988) 13. : The method of averaging and walks in inhomogeneous environments.

E. before time bk/4 , for otherwise Z 1 ∩ Z 2 = ∅. Therefore, PGW ((A1k )c ∩ Ck ) ≤ E GW PTo (X ·2 visits Z 1 before time bk/4 ) ≤ bk/4 E GW max PTo (X ·2 visits v before time bk/4 ). 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 .

J=0 (75) We proceed by estimating the latter probability. For j ≥ 1, let A1, j,n = {there exists a t ≤ jn 10 such that d X t ≥ (log jn 10 )2 }. Note that by the assumption β k pk < ∞ for some β > 1, there exists a constant c such that for all j and all n large, o PGW (A1, j,n ) ≤ e−c(log( jn 123 10 ))2 ≤ e−c(log n 10 )2 −c(log j)2 , (76) A central limit theorem for biased random walks on Galton–Watson trees 625 We next recall that t is a fresh time for the random walk if X s = X t for all s < t.

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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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