### Abstract

Let Y_{k} (ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z (ω) := ∑_{k≥0} Y_{k}(ω) is the total progeny of w. In this paper, we will prove various statistical properties of Z and Y_{k}. 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 y_{k}(ω) := ∑ _{j=0}^{k} Y_{j} (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 {Y_{k}}_{k}, and show that, as n → ∞, {Y_{k}/n}_{k} is asymptotically Gaussian under the conditional distribution P(· Z = n).

Original language | English |
---|---|

Pages (from-to) | 229-264 |

Number of pages | 36 |

Journal | Advances in Applied Probability |

Volume | 37 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2005 Mar |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

**On the number of vertices with a given degree in a Galton-Watson tree.** / Minami, Nariyuki.

Research output: Contribution to journal › Article

*Advances in Applied Probability*, vol. 37, no. 1, pp. 229-264. https://doi.org/10.1239/aap/1113402407

}

TY - JOUR

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

AU - Minami, Nariyuki

PY - 2005/3

Y1 - 2005/3

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

UR - http://www.scopus.com/inward/record.url?scp=17744382816&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17744382816&partnerID=8YFLogxK

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 -