AI 中文总结
研究任意采样图上的不完全U统计量，通过允许随机向量非同分布、核不对称且依赖边以及由任意图指定采样结构来推广。利用图论边着色结果推导\(U - \mathbb{E}U\)的集中不等式，方法简单透明且可适应更广泛设置。
AI 中文摘要
设\(X_1, X_2, \ldots, X_n\)为独立随机向量。对于顶点集\(V = \{1, 2, \ldots, n\}\)的有向图\(G=(V,E)\)和二元核集合\(\{h_e:e\in E\}\)，考虑\(U=\sum_{e=(i,j)\in E} h_e(X_i,X_j)\)。该框架通过允许随机向量非同分布、核不对称且依赖边以及由任意图指定采样结构来推广不完全U统计量。推导了\(U - \mathbb{E}U\)的几个集中不等式。主要证明策略利用图论的边着色结果，并将\(U\)的尾行为与\(G\)的色指数相关联。此方法简单、透明且易于适应更广泛的设置，包括\(m>2\)阶的U统计量和涉及双索引随机向量的统计量。
英文摘要
Let $X_1, X_2, \ldots, X_n$ be independent random vectors. For a directed graph $G=(V,E)$ with vertex set $V=\{1,2,\ldots,n\}$ and a collection of bivariate kernels $\{h_e:e\in E\}$, we consider \[ U=\sum_{e=(i,j)\in E} h_e(X_i,X_j). \] This framework generalizes incomplete U-statistics by allowing the random vectors to be non-identically distributed, the kernels to be asymmetric and edge-dependent, and the sampling structure to be specified by an arbitrary graph. We derive several concentration inequalities for $U-\mathbb{E}U$. The main proof strategy exploits edge-coloring results from graph theory and relates the tail behavior of $U$ to the chromatic index of $G$. This approach is elementary, transparent, and readily adaptable to broader settings, including U-statistics of order $m>2$ and statistics involving doubly indexed random vectors.
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