Consistent Minimal Displacement of Branching Random Walks

Abstract

Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v{\}}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\Lambda (\cdot )$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by ${\mathbb{D}}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be 0. With $\bar S_v=\max\{S_w:{\rm $w$isonthegeodesicconnectingtherootto$v$}\}$, define $L_n={\min }_{v\in {\mathbb{D}}_n}{\bar{S}}_v$. We prove that $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.

References in corpus (1)

• Almost sure convergence for stochastically biased random walks on trees