AI 中文总结
研究平稳马尔可夫链中依赖固定连续状态有限块的加性泛函，引入希尔伯特空间分解，分离不同块信息，揭示马尔可夫相关性下特征值情况及可逆时协方差结构，通过具体模型说明理论。
AI 中文摘要
我们研究平稳马尔可夫链的加性泛函，其可观测量依赖于固定的连续状态有限块。这种块可观测量自然出现在滑动窗口统计、模式计数和局部相关性分析中。在独立设置下，重叠有限块的加性泛函已知具有整数谱\(0,1,\ldots,k\)的协方差算子，且特征值为一的分量表示在块长度\(k\)首次可检测到的信息。我们研究基础序列为平稳马尔可夫链时的相应问题。对于固定的块可观测量\(f(X_t,\ldots,X_{t + k - 1})\)，我们引入一种希尔伯特空间分解，将较短连续块中已包含的信息与真正新的块\(k\)分量（称为增量分量）分开。我们表明，在马尔可夫相关性下，该分量作为特征值为一的分量持续存在：在这个空间上，格林 - 库博协方差平凡收敛，且协方差算子作用为恒等算子。更一般地，整数\(1,\ldots,k - 1\)被证明作为格林 - 库博算子的特征值分层出现。在可逆情况下，其余协方差结构通过由基础转移算子确定的边界校正单坐标边际谱来描述。可逆两态链和高斯 AR(1) 模型通过具体谱公式说明了该理论。
英文摘要
We study additive functionals of stationary Markov chains whose observables depend on a fixed finite block of consecutive states. Such block observables arise naturally in sliding-window statistics, pattern counts, and local dependence analysis. In the independent setting, additive functionals of overlapping finite blocks are known to have a covariance operator with integer spectrum \(0,1,\ldots,k\), and the eigenvalue-one component represents the information first detectable at block length \(k\). We study the corresponding problem when the underlying sequence is a stationary Markov chain. For a fixed block observable \(f(X_t,\ldots,X_{t+k-1})\), we introduce a Hilbert-space decomposition that separates information already contained in shorter consecutive blocks from the genuinely new block-\(k\) component, called the incremental component. We show that this component persists as an eigenvalue-one component under Markovian dependence: on this space the Green--Kubo covariance converges trivially and the covariance operator acts as the identity. More generally, the integers \(1,\ldots,k-1\) are shown to arise hierarchically as eigenvalues of the Green--Kubo operator. In the reversible case, the remaining covariance structure is described through a boundary-corrected one-coordinate marginal spectrum determined by the base transition operator. Reversible two-state chains and Gaussian AR(1) models illustrate the theory through concrete spectral formulas.
Comments24 pages
