Abstract:
The main objective of this survey is to present the ‘state of the art’ of those parts of the theory of independent functions which are related to the geometry of function spaces. The ‘size’ of a sum of independent functions is estimated in terms of classical moments and also in terms of general symmetric function norms. The exposition is centred on the Rosenthal inequalities and their various generalizations and sharp conditions under which the latter hold. The crucial tool here is the recently developed construction of the Kruglov operator. The survey also provides a number of applications to the geometry of Banach spaces. In particular, variants of the classical Khintchine–Maurey inequalities, isomorphisms between symmetric spaces on a finite interval and on the semi-axis, and a description of the class of symmetric spaces with any sequence of symmetrically and identically distributed independent random variables spanning a Hilbert subspace are considered.
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This publication is cited in the following 22 articles:
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S. V. Astashkin, “Sequences of independent functions and structure of rearrangement invariant spaces”, Russian Math. Surveys, 79:3 (2024), 375–457
S. V. Astashkin, “Ob izomorfnykh vlozheniyakh v klasse diz'yunktno odnorodnykh perestanovochno invariantnykh prostranstv”, Sib. matem. zhurn., 65:3 (2024), 435–445
S. V. Astashkin, “On Isomorphic Embeddings in the Class of Disjointly Homogeneous Rearrangement Invariant Spaces”, Sib Math J, 65:3 (2024), 505
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Jiao Y., Sukochev F., Xie G., Zanin D., “Rosenthal'S Inequalities: Delta-Norms and Quasi-Banach Symmetric Sequence Spaces”, Studia Math., 255:1 (2020), 55–81
Jiao Y., Sukochev F., Zanin D., “Sums of Independent and Freely Independent Identically Distributed Random Variables”, Studia Math., 251:3 (2020), 289–315
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Jiao Y., Sukochev F., Xie G., Zanin D., “Φ-moment inequalities for independent and freely independent random variables”, J. Funct. Anal., 270:12 (2016), 4558–4596
Astashkin S.V., Sukochev F.A., “Randomized Operators on n×n n × n Matrices and Applications”, Integr. Equ. Oper. Theory, 86:3 (2016), 333–358
S. V. Astashkin, “Martingale transforms of a Rademacher sequence in symmetric spaces”, St. Petersburg Math. J., 27:2 (2016), 191–206
S. V. Astashkin, F. A. Sukochev, “Orlicz sequence spaces spanned by identically distributed independent random variables in Lp-spaces”, J. Math. Anal. Appl., 413:1 (2014), 1–19
S. V. Astashkin, “Approximation of Subspaces of Symmetric Spaces Generated by Independent Functions”, Math. Notes, 96:5 (2014), 625–633
S. V. Astashkin, “Martingale Rosenthal inequalities in symmetric spaces”, Sb. Math., 205:12 (2014), 1720–1740
S. Astashkin, F. A. Sukochev, D. Zanin, “Disjointification inequalities in symmetric quasi-Banach spaces and their applications”, Pacific J. Math., 270:2 (2014), 257–285
S. V. Astashkin, “On Complementability of Subspaces in Symmetric Spaces with the Kruglov Property”, Funct. Anal. Appl., 47:2 (2013), 148–151
S. V. Astashkin, “On subspaces generated by independent functions in symmetric spaces with Kruglov property”, St. Petersburg Math. J., 25:4 (2014), 513–527
S. V. Astashkin, “Rosenthal type inequalities for martingales in symmetric spaces”, Russian Math. (Iz. VUZ), 56:11 (2012), 52–57
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