By Peres Y., Zeitouni O.

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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.

Therefore, ∞ o E GW ((τ2 − τ1 )k ) ≤ c n=1 ⎛ e−n/c n 10k ⎝ ∞ ⎞1/2 o (Tn > jn 10 ; To = ∞)⎠ ( j + 1)2k PGW . 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.

Concluding remark Throughout the paper, we have assumed that p0 = 0 and that the offspring distribution of the GW tree has exponential moments. We believe that the main results of the paper hold under weaker assumptions (when the tree is conditioned on non-extinction if p0 > 0), however proving this would require substantial further work. 123 A central limit theorem for biased random walks on Galton–Watson trees 629 Acknowledgments We thank Nina Gantert for asking the question that led to this work, and for many useful discussions.