By Peres Y., Zeitouni O.
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Extra info for A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees
Let o ∈ D1 be an arbitrary offspring of the root. 25)], the law of τ2 − τ1 under GW is identical to the law of τ1 for the walk started at v, under the measure GWv (·|To = ∞). Therefore, o o ((τ2 − τ1 )k ) = E GW (τ1k |To = ∞) = E GW o (τ k ; T = ∞) E GW o 1 o PGW (To = ∞) o (T = ∞) = P o (T = ∞). Thus, with where in the last equality we used that PGW o GW o c denoting a deterministic constant whose value may change from line to line, ∞ o ((τ2 − τ1 )k ) ≤ c E GW o (τ1k ; |X τ1 | = n, To = ∞) E GW n=1 ∞ o (Tnk ; |X τ1 | = n, To = ∞) E GW =c n=1 ∞ ≤c o o (Tn2k ; To = ∞)1/2 PGW (|X τ1 | = n)1/2 E GW n=1 ∞ ≤c o (Tn2k ; To = ∞)1/2 , e−n/c E GW n=1 where the last inequality is due to the above mentioned exponential moments on |X τ1 |.
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A Central Limit Theorem for Biased Random Walks on Galton-Watson Trees by Peres Y., Zeitouni O.