Accumulated-Variance Functionals of Hidden Markov and Semi-Markov Processes: Finite-Sample Upper and Lower Bounds
Abstract
We study two accumulated variance functionals for stationary two-state hidden Markov or hidden semi-Markov process, observed through a single trajectory, and separate identifying these observable quantities from recovering the latent parameters behind them. For bounded durations, inverting block moments gives finite-sample upper bounds, worked out with fully explicit constants for a capped-geometric Gaussian family; on fixed interior classes, the corresponding two-point lower bounds have the same rate. On prescribed local persistence cells a long-lag estimator matches the regular and saturated regimes-though only cell by cell, with a lag built around the cell's own scale, not through any single estimator that adapts across unknown scales. Unbounded durations known only through two moments are a separate matter: no uniform exponential bound is claimed there either.
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BibTeX
@misc{lombardo2026accumulatedvariance,
title = {{Accumulated-Variance Functionals of Hidden Markov and Semi-Markov Processes: Finite-Sample Upper and Lower Bounds}},
author = {Wilson Andres Lombardo},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05736475v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Accumulated-Variance Functionals of Hidden Markov and Semi-Markov Processes: Finite-Sample Upper and Lower Bounds
AU - Wilson Andres Lombardo
PY - 2026
DA - 2026/09/02
PB - HAL
UR - https://hal.science/hal-05736475v1
AB - We study two accumulated variance functionals for stationary two-state hidden Markov or hidden semi-Markov process, observed through a single trajectory, and separate identifying these observable quantities from recovering the latent parameters behind them. For bounded durations, inverting block moments gives finite-sample upper bounds, worked out with fully explicit constants for a capped-geometric Gaussian family; on fixed interior classes, the corresponding two-point lower bounds have the same rate. On prescribed local persistence cells a long-lag estimator matches the regular and saturated regimes-though only cell by cell, with a lag built around the cell's own scale, not through any single estimator that adapts across unknown scales. Unbounded durations known only through two moments are a separate matter: no uniform exponential bound is claimed there either.
ER -