论文标题
有条件的加尔顿 - 瓦特森树：形状功能性，以及子树大小及其平均值的力量总和
Conditioned Galton-Watson trees: The shape functional, and more on the sum of powers of subtree sizes and its mean
论文作者
论文摘要
对于一个复杂的数字$α$，我们考虑了加尔顿（Watson Trees）的$α$ th尺寸的$α$ th键的总和。该功能$ x_n（α）$的限制分布已确定为$ \ reta \ neq 0 $，揭示了$ \ rereα<0 $的复杂正常限制分布与$ \ re reta> 0 $的非正常限制分布之间的过渡。在本文中，我们通过证明正常的限制分布以及力矩收敛来完成图片，在缺失的情况下$ \ reα= 0 $。在所谓的形状函数$ x_n'（0）$的情况下，也建立了相同的结果，这是所有子树大小的对数的总和；这些结果是在特殊情况下较早获得的。此外，我们证明了在此情况下$ \ reta <0 $的所有瞬间的融合，此时此结果先前丢失了，并为实际$α<1/2 $的渐近平均值建立了新的结果。 $ \ reta = 0 $的一个新颖功能是，我们发现几个$α$与独立限制的联合收敛，与$ \ rereα\ neq0 $相比，该限制是$α$的连续函数。与情况的另一个区别$ \reα\ neq0 $是，当$ \ reα= 0 $时，渐近方差存在对数因素；这也适用于形状功能。 证明很大程度上基于生成功能的奇异性分析。
For a complex number $α$, we consider the sum of the $α$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(α)$ have been determined for $\Reα\neq 0$, revealing a transition between a complex normal limiting distribution for $\Reα< 0$ and a non-normal limiting distribution for $\Reα> 0$. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case $\Reα= 0$. The same results are also established in the case of the so-called shape functional $X_n'(0)$, which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. Additionally, we prove convergence of all moments in the case $\Reα< 0$, where this result was previously missing, and establish new results about the asymptotic mean for real $α< 1/2$. A novel feature for $\Reα=0$ is that we find joint convergence for several $α$ to independent limits, in contrast to the cases $\Reα\neq0$, where the limit is known to be a continuous function of $α$. Another difference from the case $\Reα\neq0$ is that there is a logarithmic factor in the asymptotic variance when $\Reα=0$; this holds also for the shape functional. The proofs are largely based on singularity analysis of generating functions.
