Weak convergence of the number of vertices at intermediate levels of random recursive trees. (16th January 2019)

• Record Type: Journal Article
• Authors: Iksanov, Alexander; Kabluchko, Zakhar
• Is Part Of: Journal of applied probability. Volume 55:Number 4(2018)
• Page Start: 1131
• Page End: 1142
• Publication Date: 2019-01-16
• Subjects: Crump–Mode–Jagers branching process, -- Gaussian process, -- intermediate level, -- random recursive tree, -- weak convergence; Primary 60F05, -- 60J80, -- Secondary 60G50, -- 60C05; 519.2
• DOI: 10.1017/jpr.2018.75
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: Let X n ( k ) be the number of vertices at level k in a random recursive tree with n +1 vertices. We are interested in the asymptotic behavior of X n ( k ) for intermediate levels k = k n satisfying k n →∞ and k n = o (log n ) as n →∞. In particular, we prove weak convergence of finite-dimensional distributions for the process ( X n ([ k n u ])) u >0, properly normalized and centered, as n →∞. The limit is a centered Gaussian process with covariance ( u, v )↦( u + v ) −1 . One-dimensional distributional convergence of X n ( k n ), properly normalized and centered, was obtained with the help of analytic tools by Fuchs et al. (2006). In contrast, our proofs, which are probabilistic in nature, exploit a connection of our model with certain Crump–Mode–Jagers branching processes.