A1675
Title: Limit laws for tree-indexed autoregression
Authors: Anand Vidyashankar - George Mason University (United States) [presenting]
Giacomo Francisci - University of Ulm (Germany)
Abstract: A branching random walk model is considered, incorporating an autoregressive structure and the point processes representing the positions. The process starts at time $0$ with a single ancestor at the origin and evolves as follows: each individual produces a random number of children whose positions $Z_{v}$ are displaced from their parents' position $Z_{Av}$ by a factor of $\rho \in \mathbb{R}$, that is, $Z_{v}=\epsilon_{v}+\rho Z_{Av}$ where $Av$ is the parent of $v$. Here, $\{ \sum_{v} \delta_{\epsilon_{v}} \}$ is a collection of i.i.d. point processes and $\delta_{x}$ is the Dirac measure at $x$. The case $\rho=1$ corresponds to the classical branching random walk. The model exhibits substantially different behavior depending on whether $|{\rho}|$ is smaller or larger than one. In both cases, the convergence of the rescaled Laplace transform of positions at generation $n$ is studied, and an analog of the Kesten-Stigum theory is established for these processes. Almost sure convergence of the rescaled average positions are also established at generation $n$, and the related central limit theorems is derived.
