05-89 Nariyuki MINAMI
On the number of vertices with a given degree in a Galton-Watson tree (105K, LaTeX 2e) Mar 1, 05
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Abstract. Let $Y_k(\omega)$ ($k\geq0$) be the number of vertices of a Galton-Watson tree $\omega$ having $k$ children, so that $Z(\omega):=\sum_{k\geq0}Y_k(\omega)$ is the total progeny of $\omega$. In this paper, we shall prove various statistical properties of $Z$ and $Y_k$. We first show, under a mild condition, an asymptotic expansion of $P(Z=n)$ as $n\to\infty$, improving the theorem of Otter (1949). Next, we show that ${\cal Y}_k(\omega):=\sum_{j=0}^kY_j(\omega)$ is the total progeny of a new Galton-Watson tree which is hidden in the original tree $\omega$. We then proceed to study the joint probability distribution of $Z$ and $\{Y_k\}_k$, and show that as $n\to\infty$, $\{Y_k/n\}_k$ is asymptotically Gaussian under the conditional distribution $P(\cdot\vert Z=n)$.

Files: 05-89.src( 05-89.keywords , vertices.tex )