Block-hierarchical covariance decompositions for finite-block additive functionals

Abstract

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.