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Sparse Rademacher chaos in symmetric spaces


Authors: S. V. Astashkin and K. V. Lykov
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 28 (2016), nomer 1.
Journal: St. Petersburg Math. J. 28 (2017), 1-20
MSC (2010): Primary 46B09
DOI: https://doi.org/10.1090/spmj/1436
Published electronically: November 30, 2016
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Abstract: A dispersed Rademacher chaos whose combinatorial dimension equals its order $ d$ is treated. It is proved that its unconditionality in a symmetric space $ X$ guarantees the equivalence of this chaos in $ X$ to the canonical basis of $ \ell _2$. In its turn, the latter property occurs if and only if $ X\supset G_{2/d}$, where $ G_{2/d}$ is the separable part of the Orlicz space $ \mathrm {ExpL}^{2/d}$ corresponding to the function $ M(u)\sim \mathrm {exp}(u^{2/d})$. Furthermore, it is shown that a chaos of an arbitrary order constructed by an arbitrary system of stochastically independent symmetric random variables is a basic sequence in any ambient symmetric space.


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Additional Information

S. V. Astashkin
Affiliation: Department of Mathematics and Mechanics, Samara State University, ul. Akademika Pavlova 1, 443011 Samara – and – Samara State Aerospace University, Moskovskoe shosse 34, 443086 Samara, Russia
Email: astash@samsu.ru

K. V. Lykov
Affiliation: Image Processing Systems Institute, RAS, ul. Molodogvardeiskaya 151, 443001 Samara – and – Samara State Aerospace University, Moskovskoe shosse 34, 443086 Samara, Russia
Email: alkv@list.ru

DOI: https://doi.org/10.1090/spmj/1436
Keywords: Rademacher functions, Khinchin inequality, Rodin--Semenov theorem, independent functions, symmetric space, unconditional basic sequence, Rademacher chaos, combinatorial dimension
Received by editor(s): April 1, 2015
Published electronically: November 30, 2016
Additional Notes: The research of the first author was supported by the Ministry of Education and Science of Russia in the framework of the basic part of the state assignment for SamGU (project no. 204). Both authors were supported by the Ministry of Education and Science of Russia within the Program of boosting the competitive ability of SGAU among world-leading scientific and educational centers (agreement no. 02.b49.21.0005). The second author was also supported by RFBR (grant no. 14-01-31452-mol-a)
Dedicated: Dedicated to Evgenii Mikhailovich Semenov on the occasion of his 75th birthday
Article copyright: © Copyright 2016 American Mathematical Society