Abstract
In this thesis, we study the limiting behavior of bootstrap random walk and oncereinforced random walk on regular tree. They are two examples of the random walk with dependent increments. In particular, we prove the invariance principle for the 2K + 1-dimensional bootstrap random walk and compute the covariance matrix for the limiting Brownian motion. Moreover, by considering the Markovian property for the increment process, we obtain a recurrence/transience analysis. For the second topic, we actually consider the MAD random walk and treat once-reinforced random walk as a special case. We illustrate a coupling between the MAD walk and the underlying environment, and extend this coupling to certain subtrees. With the inspiration from other people's work, we provide a weak large deviation principle for the local times and, as a byproduct, we prove the invariance principle for the distance of MAD random walk from the root.
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