Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

A weighted recursive tree is an evolving in which vertices are assigned random vertex-weights and new connect to a predecessor with probability proportional its weight. Here, we study the maximum degree near-maximum degrees trees when almost surely bounded their distribution function satisfies mild regularity condition near zero. We able specify higher-order corrections first order growth of established prior work. The accuracy results depends on behaviour weight largest possible value certain cases manage find up order. Additionally, describe tail degree, number attaining establish asymptotic normality degree. Our analysis extends proved for (where weights constant) case weights. main technical result shows that several uniformly chosen asymptotically independent explicit error corrections.