 0589 Nariyuki MINAMI
 On the number of vertices with a given degree in a GaltonWatson tree
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Mar 1, 05

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Abstract. Let $Y_k(\omega)$ ($k\geq0$) be the number of vertices of a
GaltonWatson 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 GaltonWatson 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 $P(\cdot\vert Z=n)$.
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