bibkey: liu2014hausdorff authors: Jian-Guo Liu; Robert L. Pego year: 2014 title: “On generating functions of Hausdorff moment sequences” doi: null url: https://arxiv.org/abs/1401.8052v4 claim: “The introductory Hausdorff characterization identifies completely monotone sequences with moments of finite positive measures on [0,1].” strata_touched: [] license: citation-only triage: anchor

Compact moment data and retention estimation

Source: https://arxiv.org/abs/1401.8052v4, §1, equation (1) and the paragraph beginning “A classical theorem of Hausdorff”. The version is v4, including its corrigendum. The consumed statement is the classical introductory characterization, not the paper’s Pick-function theorem, Lemma 3 or its corrected proof.

For a finite positive Borel measure on $[0,1]$, the moment sequence obeys ${\sum }_{j=0}^k(-1{)}^j(\genfrac{}{}{0ex}{}{k}{j})c_{n+j}\ge 0$ for all $n,k\ge 0$; conversely these inequalities characterize such sequences. The normalization $c_0=1$ makes the measure a probability measure.

LocalClock, §十四, 定理26 uses this moment interpretation for the retained-parameter channel experiment. Its finite-horizon construction and five-mode fibre are additional project data. Compact-support uniqueness of the entire moment sequence is used there through polynomial density; a finite prefix does not supply that uniqueness. No physical sampling mechanism or extra success-readout instrument is supplied by this reference.