Renewal structure of the Tree Builder Random Walk

Abstract: In this paper, we study a class of random walks that build their own tree. At each step, the walker attaches a random number of leaves to its current position. The model can be seen as a subclass of the Random Walk in Changing Environments (RWCE) introduced by G. Amir, I. Benjamini, O. Gurel-Gurevich and G. Kozma. We develop a renewal framework for the process analogous to that established by A-S. Sznitman and M. Zerner in the context of RWRE. This provides a more robust foundation for analyzing the model. As a result of our renewal framework, we estabilish several limit theorems for the walker's distance, which include the Strong Law of Large Numbers (SLLN), the Law of the Iterated Logarithm (LIL), and the Invariance Principle, under an i.i.d. hypothesis for the walker's leaf-adding mechanism. Further, we show that the limit speed defined by the SLLN is a continuous function over the space of probability distributions on $\mathbb{N}$.