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The Gaussian entropy map in valued fields

Yassine El Maazouz

Vol. 13 (2022), No. 1, 1–18

The entropy map for multivariate real valued Gaussian distributions is the map that sends a positive definite matrix Σ to the sequence of logarithms of its principal minors (log (det ΣI))I. We exhibit the analog of this map in the nonarchimedean local fields setting, like the field of p-adic numbers for example. As in the real case, the image of this map lies in the supermodular cone. Moreover, given a multivariate Gaussian measure on a local field, its image under the entropy map determines its pushforward under valuation. In general, this map can be defined for nonarchimedean valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3.

entropy, probability, Gaussian measures, nonarchimedean valuation, local fields, Bruhat–Tits building, conditional independence
Mathematical Subject Classification
Primary: 12J25, 60E05, 94A17
Received: 3 January 2021
Revised: 18 January 2022
Accepted: 10 March 2022
Published: 4 December 2022
Yassine El Maazouz
Department of Statistics
University of California
Berkeley, CA
United States