滑动窗口占用计数的谱间隙的最优状态空间阶数
Optimal State-Space Order for Spectral Gaps of Sliding-Window Occupation Counts
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中文总结 AI 辅助
本文研究了滑动窗口占用计数核的谱间隙与原始马尔可夫核谱间隙的最优比较系数，证明了其阶为 m 的倒数，并给出了上下界。
中文摘要 AI 辅助
设 $P$ 是定义在 $m$ 状态空间 $\Omega$ 上的不可约可逆马尔可夫核，其右谱间隙记为 $\gamma=1-\lambda_2(P)$。从平稳轨迹出发，令 $K_t$ 为从时间 $t$ 开始的长度为 $n$ 的窗口的占用计数向量。平稳对 $(K_0,K_1)$ 定义了一个可逆投影计数核 $\widetilde P_n$。对于每个 $m\ge2$，定义 \\[ c_m^\star = \inf_{\substack{ P,\\; n \ge 2}} \frac{n\Gap(\widetilde P_n)}{\Gap(P)}. \\] 我们证明 \\[ \frac1{1080m}\le c_m^\star\le q_{m-2}, \qquad q_0=\frac14,\quad q_{r+1}=q_r(1-q_r). \\] 我们还证明了 $q_{m-2}=(m+\log m+O(1))^{-1}$，这意味着 $c_m^\star=\Theta(m^{-1})$。因此，最优比较系数的阶为 $m$，但其精确值仍然开放。下界对 $n=1$ 也成立，并且对 $P$ 一致成立，包括稀疏核和周期核。其证明结合了短窗口去相关与平均锚点-游走分解、格林核击中估计以及停止的 Carleson-Hardy 不等式。一个嵌套稀有状态构造产生了有限 $m$ 状态见证者，其归一化瑞利商通过有序极限序列逼近 $q_{m-2}$。对于每个固定的有限不可约可逆非周期核（至少两个状态），$\Gap(\widetilde P_n)=\Theta_P(n^{-1})$。
英文摘要
Let $P$ be an irreducible reversible Markov kernel on a $m$-state space $Ω$, and denote its right spectral gap $γ=1-λ_2(P)$. From a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible projected count kernel $\widetilde P_n$. For every $m\ge2$, let \[ c_m^\star= \inf_{\substack{ P,\; n \ge 2}} \frac{n\Gap(\widetilde P_n)}{\Gap(P)}. \] We prove \[ \frac1{1080m}\le c_m^\star\le q_{m-2}, \qquad q_0=\frac14,\quad q_{r+1}=q_r(1-q_r). \] We also show $q_{m-2}=(m+\log m+O(1))^{-1}$, which implies that $c_m^\star=Θ(m^{-1})$. Thus, the optimal comparison coefficient has order $m$, although its exact value remains open. The lower bound also holds for $n=1$ and is uniform in $P$, including sparse and periodic kernels. Its proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson--Hardy inequality. A nested rare-state construction produces finite $m$-state witnesses whose normalized Rayleigh quotients approach $q_{m-2}$ through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, $\Gap(\widetilde P_n)=Θ_P(n^{-1})$.
发表机构
- School of Mathematical Sciences, Peking University（北京大学数学科学学院）
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