On the number of vertices with a given degree in a Galton-Watson tree

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Abstract

Let Yk (ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z (ω) := ∑k≥0 Yk(ω) is the total progeny of w. In this paper, we will prove various statistical properties of Z and Yk. We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that yk(ω) := ∑ j=0k Yj (ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree w. We then proceed to study the joint probability distribution of Z and {Yk}k, and show that, as n → ∞, {Yk/n}k is asymptotically Gaussian under the conditional distribution P(· Z = n).

Original languageEnglish
Pages (from-to)229-264
Number of pages36
JournalAdvances in Applied Probability
Volume37
Issue number1
DOIs
Publication statusPublished - 2005 Mar 1

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Keywords

  • Central limit theorem
  • Galton-Watson tree
  • Lagrange inversion
  • Total progeny

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics

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