Random walks on Complete Binary Trees
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05-11-2019 - |
سؤال
Let $T$ be a complete binary tree of height $n$ and root $r$.
A random walk starts at $r$, and at each step uniformly at random moves on a neighbor.
There are $m$ random walkers all starting at $r$ and let denote with $H_1,\dots,H_m$, the heights reached by the walkers after $n$ steps.
Show that, for some constant $C$ which do not have to depend on $n$ and $m$, it holds that
$\mathbb{P}\left(\underset{i \in [m]}\max\left|H_i - \frac{n}{3}\right| \le C \sqrt{n\ln m}\right) \ge 1 - \frac{1}{m}$
I have been trying several strategies, to appropriately define $H_i$ as sum of random variables and similar, but no one turned out to work. Do you have any idea/suggestion to attach this problem?
Thanks in advance!
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