论文标题
对于统一的奇异马尔可夫链的U级统计的浓度不平等
Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains
论文作者
论文摘要
对于统一的奇异马尔可夫链的U级统计，我们证明了新的浓度不平等。使用有界和$π$的核心内核，我们表明我们可以恢复Arcones和Gin {é}的收敛速率，这些弧菌证明了独立随机变量和规范内核的U统计量的浓度。我们的结果允许将内核$ h_ {i，j} $与索引中的索引相关联，从而阻止了使用标准阻止工具的使用。我们的证明依赖于归纳分析，在该分析中，我们使用Martingale技术，均匀的麦芽糖分裂和伯恩斯坦的类型不平等。假设马尔可夫链从其不变分布开始，我们证明了伯恩斯坦型浓度不平等，可为小方差项提供更清晰的收敛速度。
We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $π$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{é} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein's type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.
