By Peres Y., Zeitouni O.

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**Additional resources for A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees**

**Sample text**

Thus, o (Tn > jn 10 , To = ∞, Ac2, j,n ) ≤ (1 − c) j PGW 1/2 n 3 . (78) Combining (77) and (78), we conclude that ∞ o (Tn > jn 10 , To = ∞) ≤ c. ( j + 1)2k PGW j=0 Substituting in (75), the lemma follows. g. 24]) is the following: Corollary 4 There exists a√constant σ 2 such that, under the annealed measure GW, the process {(|X nt | − nvt)/ σ 2 n}t≥0 converges in distribution to a Brownian motion. Proof of Theorem 3 Our argument is based on the technique introduced by Bolthausen √ and Sznitman [7], as developed in [8].

Zeitouni n . Consider the and let Bnt (|X |· ) denote the polygonal interpolation of (k/n) → Bk/n space C T of continuous functions on [0, T ], endowed with the distance dT (u, u ) = supt≤T |u(t) − u (t)| ∧ 1. 1], Theorem 3 will follow from Corollary 4 once we show that for all bounded by 1 Lipschitz function F on C T with Lipschitz constant 1, and b ∈ (1, 2], varGW E To [F(B bk )] < ∞. (79) k In the sequel, fix b and F as above. For the same tree T , let X ·1 and X ·2 be independent λ-biased random walks on T , and set B[i, k]t = Bt Bt bk (|X i | ·+s − |X i | s ), bk (|X i |· ) and B[i, k, s]t = i = 1, 2.

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.

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

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