Let $\Gamma\_0,\Gamma\_1,...$ be regeneration epochs.


If $(X\_n)\_{n \in \mathbb{N}}$ is a $\lambda$ biased random walk on a Galton-Watson tree, than the regeneration epochs are defined as:


$\Gamma\_0:=\inf\{\iota \ | X\_i\neq X\_{\iota} \ \forall i\leq \iota \ \text{and} \ X\_{j}\neq (X\_{\iota})\_\* \ \forall j\geq \iota \}$,


$\Gamma\_k:=\inf\{\iota \ | \iota>\Gamma\_{k-1}: X\_i\neq X\_{\iota} \ \forall i\leq \iota \ \text{and} \ X\_{j}\neq (X\_{\iota})\_\* \ \forall j\geq \iota \}$




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I want to show that


$\{\Gamma\_0 < \infty \}=\mathcal{S}$, where $\mathcal{S}$ is the survival set.




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The statement makes sense, of course. But i do not know how to start the proof.


Maybe someone has experience in the subject area and could help me.


Greetings : Fynn



Random walks on GW-trees (regeneration epochs/survival set)
You need to assume that the bias satisfies $\lambda<m$ where $m$ is the mean offspring. Then you can find the proof in Lemma 3.3 page 253 of [1].


[1] Lyons, Russell, Robin Pemantle, and Yuval Peres. "Biased random walks on Galton–Watson trees." Probability theory and related fields 106, no. 2 (1996): 249-264.


<https://link.springer.com/content/pdf/10.1007/s004400050064.pdf>


