条件重采样滑动窗口计数核：谱隙界与庞加莱不等式
Conditionally Resampled Sliding-Window Count Kernels: Spectral-Gap Bounds and Poincaré Inequalities
AI总结：
该研究针对平稳有限状态可逆马尔可夫链的条件重采样滑动窗口计数核，推导了其谱隙的Θ_P(1/n)界及庞加莱不等式，得到有限窗口计数统计量的方差界与矩阵值经验平均的算子范数集中性。
AI中文摘要：
我们研究与平稳有限状态可逆马尔可夫链的长度为n的窗口经验计数相关的条件重采样滑动窗口计数核。尽管所得计数过程通常不是马尔可夫过程，但其平稳一步条件律定义了一个真正的马尔可夫核。对于有限状态空间上的每个固定严格正可逆核P，我们为长度为n的诱导计数核P~ₙ给出了一个庞加莱不等式，即推导了P~ₙ的谱隙Gap(P~ₙ)的下界为Gap(P~ₙ)≥c(P)/n，其中c(P)>0仅依赖于P。该证明结合了平稳路径律的鞅振荡不等式与坐标振荡和计数核的狄利克雷形式的直接比较。计数向量的线性统计量给出了匹配的O(1/n)上界，因此对于每个固定严格正可逆P，有Gap(P~ₙ)=Θ_P(1/n)。所得的计数空间庞加莱不等式给出了有限窗口计数统计量的局域到全局方差界，并结合一般矩阵集中原理，得到了矩阵值经验平均的算子范数集中性。
英文摘要：
We study the conditionally resampled sliding-window count kernel associated with the empirical counts of length-$n$ windows from a stationary finite-state reversible Markov chain. Although the resulting count process is generally not Markov, its stationary one-step conditional law defines a genuine Markov kernel. For every fixed strictly positive reversible kernel \(P\) on a finite state space, we present a Poincaré inequality for the induced count kernel $\tP_n$ of length $n$. In other words, we derive the lower bound of the spectral gap $\Gap(\tP_n)$ of $\tP_n$ as \[ \Gap(\tP_n)\ge \frac{c(P)}{n}, \] where \(c(P)>0\) depends only on \(P\). The proof combines a martingale oscillation inequality for the stationary path law with a direct comparison of coordinate oscillations to the Dirichlet form of the count kernel. A linear statistic of the count vector gives the matching \(O(1/n)\) upper bound, so for every fixed strictly positive reversible \(P\) one has \(\Gap(\tP_n)=Θ_P(1/n)\). The resulting count-space Poincaré inequality yields a local-to-global variance bound for finite-window count statistics and, together with a general matrix-concentration principle, operator-norm concentration for matrix-valued empirical averages.
