Limit Theorems for the Minimal Position of a Branching Random Walk in Random Environment

#### W. Hong, W. Hou, X. Zhang

2020, v.26, Issue 5, 839-860

ABSTRACT

X.Zhang (1st author), W. Hou (2nd author), W. Hong (3rd author)

We consider a branching system of $\mathbb{N}$-valued random walks with a random environment in location. We will give the exact limit value of $\frac{M_{n}}{n}$, where $M_{n}$ denotes the minimal position of the branching random walk at time $n$. A key step in the proof is to transfer our branching random walks with a random environment in location to branching random walks with a random environment in time, by use of Bramson's ``branching processes within a branching process" (1978).

Keywords: random walk in random environment, branching random walk, branching process, minimal position

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