bibkey: liptser2001sequential
authors: Robert S. Liptser and Albert N. Shiryaev
year: 2001
title: Statistics of Random Processes
doi: 10.1007/978-3-662-13043-8
url: https://doi.org/10.1007/978-3-662-13043-8
claim: Martingale localization and conditional Hellinger processes relate finite predictable energy to likelihood convergence.
strata_touched:
  - D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/SequentialHellingerLocalization
license: reference-only
triage: anchor

Conditional energy and localization

The reference develops likelihood ratios, changes of measure, and martingale methods for stochastic processes. The sequential Hellinger criterion and the localization method are classical mathematics; no novelty is claimed here. The DOI, title, and authors are identified by the Crossref record for the linked Springer volume. No book text or source code is reproduced.

The formal statement uses a discrete predictable energy sequence with values in the unit interval. The cut at time n tests the energy accumulated before n. One included increment may cross the chosen level, so the total cut energy is bounded by that level plus one. Conditional second-moment bounds then control centered partial sums, while conditional first-moment bounds control a nonnegative error sum. Countably many integer levels give one full-measure set on which both conclusions hold on every finite-energy path.

For strictly positive normalized rows on a finite alphabet, the formalization connects this localization to the actual full-history trajectory laws. The finite-energy event carries mutually absolutely continuous restrictions; the complementary restrictions are mutually singular. Divergence of the negative-log-affinity partial sums under both laws implies global singularity. The measure bridge uses conditional expectations of global densities with respect to the sum of the two laws. This finite-alphabet scope does not supply the broader regular-kernel statement.

Verified locator

• DOI and publisher locator: https://doi.org/10.1007/978-3-662-13043-8