Abstract:
For a broad class of Banach spaces with Gaussian measure, we show that their
entropy in the sense of Shannon (the mm-entropy) is closely
related to the entropy of the corresponding ellipsoid of concentration and
behaves, in a certain range, as the logarithm of the measure of small balls.
Relations between the mm-entropy and the entropy of compact sets
are also discussed in light of the classical works of Kolmogorov and
Shannon.
Keywords:
Gaussian measure, mm-entropy, entropy of compact sets.
Citation:
A. M. Vershik, M. A. Lifshits, “On mm-entropy of a Banach space with a Gaussian measure”, Teor. Veroyatnost. i Primenen., 68:3 (2023), 532–543; Theory Probab. Appl., 68:3 (2023), 431–439
\Bibitem{VerLif23}
\by A.~M.~Vershik, M.~A.~Lifshits
\paper On $\mathrm{mm}$-entropy of a Banach space with a Gaussian measure
\jour Teor. Veroyatnost. i Primenen.
\yr 2023
\vol 68
\issue 3
\pages 532--543
\mathnet{http://mi.mathnet.ru/tvp5637}
\crossref{https://doi.org/10.4213/tvp5637}
\transl
\jour Theory Probab. Appl.
\yr 2023
\vol 68
\issue 3
\pages 431--439
\crossref{https://doi.org/10.1137/S0040585X97T991544}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85179303429}
Linking options:
https://www.mathnet.ru/eng/tvp5637
https://doi.org/10.4213/tvp5637
https://www.mathnet.ru/eng/tvp/v68/i3/p532
This publication is cited in the following 4 articles:
A. I. Bufetov, I. A. Ibragimov, M. A. Lifshits, A. V. Malyutin, F. V. Petrov, N. V. Smorodina, A. N. Shiryaev, Yu. V. Yakubovich, “In memory of A. M. Vershik (28.12.1933 – 14.02.2024)”, Theory Probab. Appl., 69:2 (2024), 331–335
A. M. Vershik, G. A. Veprev, P. B. Zatitskii, “Dynamics of metrics in measure spaces and scaling entropy”, Russian Math. Surveys, 78:3 (2023), 443–499
A. M. Vershik, “Classification of measurable functions of several variables and matrix distributions”, Funct. Anal. Appl., 57:4 (2023), 303–313
A. A. Tadevosyan, “Ob mm-entropii raspredelenii gaussovskikh protsessov”, Veroyatnost i statistika. 34, Posvyaschaetsya yubileyu Andreya Nikolaevicha BORODINA, Zap. nauchn. sem. POMI, 525, POMI, SPb., 2023, 122–133
Citation in format AMSBIB
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