By Peres Y., Zeitouni O.
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Additional info for A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees
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C ) ≤ b−c3 k . On the other hand, the event C Using (81), it follows that PGW (Ck,1 k,1 ∩ 1 1 c k/8 (Ck,1 ) implies that the sum of the difference τi+1 − τi , i = 1, . . , b , is larger 123 628 Y. Peres, O. Zeitouni than bk/4 /2, and hence, by Markov’s inequality, PGW (Ck,1 ∩ (Ck,1 )c ) ≤ 2bk/8 E GW (τ21 − τ11 ) ≤ b−c5 k , bk/4 for some deterministic constant c5 < c4 . 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 .
A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees by Peres Y., Zeitouni O.