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Abstract
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The entropy map for multivariate real valued Gaussian distributions is the map that sends a positive
definite matrix
to the sequence of logarithms of its principal minors
. We
exhibit the analog of this map in the nonarchimedean local fields setting, like the field of
-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
.
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Keywords
entropy, probability, Gaussian measures, nonarchimedean
valuation, local fields, Bruhat–Tits building, conditional
independence
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Mathematical Subject Classification
Primary: 12J25, 60E05, 94A17
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Milestones
Received: 3 January 2021
Revised: 18 January 2022
Accepted: 10 March 2022
Published: 4 December 2022
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© 2022 MSP (Mathematical Sciences
Publishers). |
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