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

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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. 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.

Let T = min{t > 0 : |X t | = }. Let Yt be a nearest neighbor random walk on Z+ with P(Yt+1 = Yt −1|Yt ) = λ/(λ+1) whenever Yt = 0. Y· and X · can be constructed on the same probability space, such that T ≤ min{t > 0 : Yt = } =: T Y for all . On the other hand, using the Markov property, for any constant c and all large, P(T Y >e )≤ 1− c 1 1+λ ec / In particular, there exists a c1 = c1 (λ) > 0 such that PGW (T > ec1 ) ≤ e− /c1 (better bounds are available but not needed). Thus, for some deterministic constants ci = ci (λ, b) > 0, i ≥ 2, and all k large, PGW (τ1 > bk/4 /2) ≤ PGW (|X τ1 | > c2 k) + PGW (τ1 > bk/4 /2, |X τ1 | ≤ c2 k) ≤ PGW (|X τ1 | > c2 k) + PGW (Tc2 k > bk/4 /2) ≤ e−c3 k , (81) where we have used the above mentioned fact that |X τ1 | possesses exponential moments.

A new approach to the limit theory of recurrent Markov chains. Trans. Am. Math. Soc. 245, 493–501 (1978) 4. : Weighted sums of certain dependent random variables. Tôhoku Math. J. 19, 357–367 (1967) 5. : Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137, 83–120 (2007) 6. : Convergence of Probability Measures, 2nd edn. Wiley, New York (1999) 7. : On the static and dynamic points of view for certain random walks in random environment. Methods Appl.

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


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