通过额外随机元素的乘积矩获取集中度
Concentration from Product Moments via an Additional Element of Randomness
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中文总结 AI 辅助
该研究在初等对称多项式框架中引入额外随机元素，将乘积矩边界限定问题应用于三类场景，改进了读-Δ族的边界依赖，推导了哈希负载边界，恢复了马尔可夫链的谱与混合时间尺度。
中文摘要 AI 辅助
用于证明集中度边界的标准指数矩方法，常可替换为基于初等对称多项式的论证。我们向该框架引入额外随机元素，将问题简化为对均匀采样索引集上的乘积矩进行边界限定。我们证明该方法在三类场景中能给出有效边界：针对有限独立性下的读-Δ（read-Δ）族，得到由随机诱导依赖子图度数决定的边界，改进了对最坏情况度数的依赖；针对含（半）随机输入的随机二元线性哈希，通过控制输入键随机元组的秩缺陷，推导了固定桶与最大负载边界；最后，针对随机过程，我们展示乘积矩的衰减如何产生集中度边界，恢复了有限状态马尔可夫链的谱与混合时间尺度。
英文摘要
The standard method of exponential moments for proving concentration bounds can often be replaced by an argument based on elementary symmetric polynomials. We introduce an additional element of randomness into this framework, which reduces the problem to bounding product moments over a uniformly sampled set of indices. We show that this approach gives useful bounds in three settings. For read-$Δ$ families under limited independence, we obtain bounds governed by the degrees of randomly induced dependency subgraphs, improving the dependence on worst-case degrees. For random binary linear hashing with (semi-)random inputs, we derive fixed-bin and maximum-load bounds by controlling the rank defect of random tuples of input keys. Finally, for stochastic processes, we show how decay of product moments yields concentration bounds, recovering the spectral and mixing-time scales for finite-state Markov chains.
