Block-hierarchical covariance decompositions for finite-block additive functionals

What the math gives to ML

The paper develops an orthogonal hierarchy for finite-window observables of stationary Markov sequences: each window feature is separated into information already present in shorter windows and an incremental component that first appears at the current window length. The key transferable asset is that overlapping-window covariance is structured by powers of a one-step transition operator, yielding a commuting Kac–Murdock–Szegő-type block Toeplitz matrix with an explicit tridiagonal inverse. This suggests an overlap-aware sequence module that residualizes local features against shorter-context predictors and applies an analytically motivated precision filter before attention. The exact guarantees require a stationary reversible Markov process, but the transition operator and projections can be estimated from minibatch sequences and tested as approximations.