By Peres Y., Zeitouni O.

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

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H. Poincare 39, 527–555 (2003) 9. : A transmutation formula for Markov chains. Bull Sci. Math. 109, 399–405 (1985) 10. : Large deviations for random walks on Galton–Watson trees: averaging and uncertainty. Probab. Theory Relat. Fields 122, 241–288 (2001) 11. : Random walks and electric networks. Carus Mathematical Monographs, vol. 22. Mathematical Association of America, Washington, DC (1984) 12. : Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Heidelberg (1988) 13. : The method of averaging and walks in inhomogeneous environments.

Where Markov’s inequality was used in the third step. 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).

The method of averaging and walks in inhomogeneous environments. Russ. Math. Surv. 40, 73–145 (1985) 14. : Random walks and percolation on trees. Ann. Probab. 18, 931–958 (1990) 15. : Probability on trees and networks. iu. html 16. : Conceptual proofs of L log L iteria for mean behavior of branching processes. 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.