Asymptotic Normality of the Hill Estimator in Linear Preferential Attachment

Abstract

We prove asymptotic normality of the Hill estimator based on the largest $k_n$ degrees in a homogeneous linear preferential attachment graph with $n$ vertices. Each new vertex sends a fixed number $m\ge1$ of edges sequentially to older vertices, with attachment weight $d+δ$ and fixed $δ>-m$. Let $α=m/(2m+δ)$. For every deterministic sequence $k_n\to\infty$ satisfying $\sqrt{k_n}(k_n/n)^α\to0$, the estimator, centered at $α$ and multiplied by $\sqrt{k_n}$, converges to a centered normal law with an explicit variance strictly smaller than $α^2$. The reduction relative to the Hill variance for independent observations separates into contributions from the different degree laws at different birth times and from competition for new edges. The contribution from competition is $1/m$ times that from different birth times. Representations using Gamma and beta functions give a consistent standard error and a studentized normal limit. The proof uses martingales indexed by edge attachments and comparisons with independent pure-birth processes, while retaining the integer degrees and their ties.