On the covariances of outdegrees in random plane recursive trees

TL;DR

This paper simplifies the expression for the limiting covariance matrix of outdegrees in random plane recursive trees, building on prior work that established their joint asymptotic normality.

Contribution

It provides a significantly simplified formula for the covariance matrix, enhancing understanding of the asymptotic behavior of outdegrees in these trees.

Findings

Explicit simplified covariance matrix derived

Asymptotic normality of outdegrees confirmed

Improved analytical tools for studying recursive trees

Abstract

In 2005 Janson, extending earlier work of Mahmoud, Smythe, and Szyma\'nski, established the joint asymptotic normality of the outdegrees of a random plane recursive tree. In particular, he gave an explicit description of the limiting covariance matrix. Our aim here is to provide substantially simplified expression for the limiting covariance matrix.