TY - JOUR

T1 - On the number of vertices with a given degree in a Galton-Watson tree

AU - Minami, Nariyuki

PY - 2005/3/1

Y1 - 2005/3/1

N2 - 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).

AB - 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).

KW - Central limit theorem

KW - Galton-Watson tree

KW - Lagrange inversion

KW - Total progeny

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U2 - 10.1239/aap/1113402407

DO - 10.1239/aap/1113402407

M3 - Article

AN - SCOPUS:17744382816

VL - 37

SP - 229

EP - 264

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 1

ER -