By Peres Y., Zeitouni O.
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Extra info for A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees
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 .
Ann. Appl. Probab. 13, 475–489 (2003) 20. : Galton–Watson trees with the same mean have the same polar sets. Ann. Probab. 23, 1102–1124 (1995) 21. : Théorème central limite fonctionnel pour une marche au hasard en environment aléatoire. Ann. Probab. 26, 1016–1040 (1998) 22. : Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Relat. Fields 129, 219–244 (2004) 23. : Slowdown estimates and central limit theorem for random walks in random environment.
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. Probab. 13, 475–489 (2003) 20. : Galton–Watson trees with the same mean have the same polar sets. Ann. Probab. 23, 1102–1124 (1995) 21.
A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees by Peres Y., Zeitouni O.