Abstract
If X n = ξ 1 ξ 2⋯ξ n is a transient random walk on a finitely generated group Γ with step distribution μ, then the probability q that it will never return to its initial location X 0 = 1 is positive. Moreover, for any \(n \in \mathbb {N}\) the sequence \((X_{n}^{-1}X_{n+m})_{m \geq 0}\) is a version of the random walk, as its increments ξ n+1, ξ n+2, ⋯ are independent and identically distributed with common distribution μ, and so the probability that it will ever return to the initial state 1 is also q > 0.
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Lalley, S. (2023). The Ergodic Theorem. In: Random Walks on Infinite Groups. Graduate Texts in Mathematics, vol 297. Springer, Cham. https://doi.org/10.1007/978-3-031-25632-5_2
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