A law of the iterated logarithm for iterated random walks, with application to random recursive trees

TL;DR

This paper establishes a law of the iterated logarithm for the number of individuals in a generation of a Crump-Mode-Jagers process generated by an increasing random walk, with applications to the structure of random recursive trees.

Contribution

It introduces a law of the iterated logarithm for a specific class of branching processes and applies it to analyze the asymptotic behavior of vertices at fixed levels in random recursive trees.

Findings

Law of the iterated logarithm for $Y_k(t)$ as $t o \infty$

Asymptotic behavior of vertices at fixed level in recursive trees

Quantitative description of growth in Crump-Mode-Jagers processes

Abstract

Consider a Crump-Mode-Jagers process generated by an increasing random walk whose increments have finite second moment. Let $Y_k(t)$ be the number of individuals in generation $k\in \mathbb N$ born in the time interval $[0,t]$. We prove a law of the iterated logarithm for $Y_k(t)$ with fixed $k$, as $t\to +\infty$. As a consequence, we derive a law of the iterated logarithm for the number of vertices at a fixed level $k$ in a random recursive tree, as the number of vertices goes to $\infty$.