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

Part of: Markov processes Limit theorems Combinatorics Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Nariyuki Minami
Nariyuki Minami*
Affiliation:
University of Tsukuba
*
∗ Postal address: Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan. Email address: [email protected]
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Abstract

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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≥0Yk(ω) is the total progeny of ω. 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=0kYj(ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree ω. We then proceed to study the joint probability distribution of Z and Ykk, and show that, as n → ∞, Yk/nk is asymptotically Gaussian under the conditional distribution P(· | Z = n).

Keywords

Galton-Watson treetotal progenyLagrange inversioncentral limit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60C05: Combinatorial probability 05A15: Exact enumeration problems, generating functions 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 229 - 264
Copyright
Copyright © Applied Probability Trust 2005 

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