多维随机在线最近邻图的渐近理论
Asymptotic theory for the multidimensional random on-line nearest-neighbour graph
摘要：
在$(0,1)^d$（$d \in \N$）中的一系列$n$均匀随机点上的在线最近邻图将第一个点之后的每个点连接到其前任中的最近邻点。对于这个图的总的幂加权边长，我们证明了具有权指数$α\in (0,d/2]$的方差的$O(\max \{n^{1-(2α/d)}, \log n \})$上界。另一方面，我们给出了$α> d/2$在总权边长度下的$n \to \infty$大样本收敛结果。当底层点集是强度为$n$的泊松过程时，我们证明了相应的结果.
英文摘要：
The on-line nearest-neighbour graph on a sequence of $n$ uniform random points in $(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent $α\in (0,d/2]$, we prove $O(\max \{n^{1-(2α/d)}, \log n \})$ upper bounds on the variance. On the other hand, we give an $n \to \infty$ large-sample convergence result for the total power-weighted edge-length when $α> d/2$. We prove corresponding results when the underlying point set is a Poisson process of intensity $n$.
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