By Peres Y., Zeitouni O.
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Extra resources for A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees
Theory Relat. Fields 129, 219–244 (2004) 23. : Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 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.
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 . On Ck there are at most bk/4 vertices in Z 1 . 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 .
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A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees by Peres Y., Zeitouni O.