Limit theorems for vertex-reinforced jump processes on regular trees

TL;DR

This paper establishes the strong law of large numbers and the central limit theorem for the distance of a vertex-reinforced jump process on regular trees with degree at least 3, advancing understanding of its long-term behavior.

Contribution

It provides the first rigorous proofs of limit theorems for vertex-reinforced jump processes on regular trees, including the case of degree at least 3.

Findings

Proved strong law of large numbers for the process.

Established central limit theorem for the process.

Identified open problem of transience on binary trees.

Abstract

Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactly $b$ children, with $b \ge 3$. We prove the strong law of large numbers and the central limit theorem for the distance of the process from the root. Notice that it is still unknown if vertex-reinforced jump process is transient on the binary tree.