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Asymptotic Joint Normality of Outdegrees of Nodes in Random Recursive Trees
Author(s) -
Mahmoud Hosam M.,
Smythe R. T.
Publication year - 1992
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.3240030305
Subject(s) - joint probability distribution , mathematics , asymptotic distribution , combinatorics , marginal distribution , multivariate normal distribution , covariance , tree (set theory) , distribution (mathematics) , limiting , random variate , normal distribution , probability distribution , joint (building) , discrete mathematics , statistics , multivariate statistics , random variable , mathematical analysis , mechanical engineering , architectural engineering , estimator , engineering
We study the joint probability distribution of the number of nodes of outdegree 0, 1, and 2 in a random recursive tree. We complete the known partial list of exact means and variances for outdegrees up to two by obtaining exact combinatorial expressions for the remaining means, variances, and covariances. The joint probability distribution of the number of nodes of outdegree 0, 1, and 2 is shown to be asymptotically trivariate normal and the asymptotic covariance structure is explicitly determined. It is also shown how to extend the results (at least in principle) to obtain a limiting multivariate normal distribution for nodes of outdegree 0, 1, …, k .