幂权递归树中的总路径长度：鞅极限与全局波动
Total Path Length in Power-Weight Recursive Trees: Martingale Limits and Global Fluctuations
- Faculty of Mathematics and Computer Science, Adam Mickiewicz University（亚当·密茨凯维奇大学数学与计算机科学系）
- Department of Statistics, Poznań University of Economics and Business（波兹南经济商业大学统计系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究在幂权递归树中建立了总路径长度的鞅极限与方差渐近，覆盖所有实数幂指数，并揭示与随机二叉搜索树在全局波动上的差异。
AI中文摘要：
我们研究了具有正确定性附着权重的递归树中的总路径长度。记 $W_n=\sum_{i=1}^n w_i$ 和 $p_n=w_n/W_n$，我们得到了精确的鞅-创新恒等式和方差递推关系。在条件 $p_n=O(n^{-1})$ 下，中心化的总路径长度除以 $n$ 几乎必然且依 $L^2$ 收敛到一个非退化随机变量，其方差渐近于一个正常数乘以 $n^2$。不需要 $W_n$ 的多项式渐近性。对于幂权重 $w_i=i^\alpha$，同样的论证适用于每个实数 $\alpha$，包括超出现有轮廓理论正幂累积权重假设的临界和可求和区域。对于 $\alpha>-1$，期望平均深度是对数阶；对于 $\alpha=-1$，是迭代对数阶；对于 $\alpha<-1$，是有界的，而全局波动尺度在整个过程中保持线性。在可求和区域中，我们通过无限树的加权深度来识别随机极限。均匀情形恢复了经典方差系数 $2-\pi^2/6$。对于线性权重，我们计算出系数为 $8-2\pi^2/3$。尽管该树和随机二叉搜索树具有相同的插入深度边际分布和期望总路径长度，但它们的渐近方差系数相差 1。这给出了在单个深度分布中不可见的全局依赖性的明确比较。
英文摘要：
We study total path length in recursive trees with positive deterministic attachment weights. Writing $W_n=\sum_{i=1}^n w_i$ and $p_n=w_n/W_n$, we obtain exact martingale-innovation identities and a variance recurrence. Under the condition $p_n=O(n^{-1})$, centered total path length divided by $n$ converges almost surely and in $L^2$ to a nondegenerate random variable, and its variance is asymptotic to a positive constant times $n^2$. No polynomial asymptotic for $W_n$ is required. For power weights $w_i=i^α$, the same argument applies to every real $α$, including the critical and summable regimes beyond the positive-power cumulative-weight assumptions of existing profile theory. The expected average depth is logarithmic for $α>-1$, iterated logarithmic for $α=-1$, and bounded for $α<-1$, while the global fluctuation scale remains linear throughout. In the summable regime we identify the random limit through the weighted depths of the infinite tree. The uniform case recovers the classical variance coefficient $2-π^2/6$. For linear weights we evaluate the coefficient as $8-2π^2/3$. Although this tree and a random binary search tree have identical insertion-depth marginals and expected total path length, their asymptotic variance coefficients differ by one. This gives an explicit comparison of global dependence that is invisible in individual depth distributions.
