Independent functions and the geometry of Banach spaces. (English. Russian original) Zbl 1219.46025

Astashkin, S. V.; Sukochev, F. A.

Russ. Math. Surv. 65, No. 6, 1003-1081 (2010).

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.
More precisely, suppose that \(\{f_k\}_{k=1}^\infty\) is a sequence of independent random variables on some probability space. It is required to somehow estimate the ‘size’ of the sums \(S_n=\sum_{k=1}^n f_k\). Whereas in probability theory, mainly the ‘tail’ distribution of the sum, that is, the quantity \(\mathbb{P}\{|S_n|>\tau\}\), is of interest, here in this Banach space approach, the norm \(\| S_n \|_X\) with respect to a given function space is investigated and estimated. First and foremost, information on the classical moments \(\| S_n \|_p=(\mathbb{E}|S_n|^p)^{1/p}\) \((p>0)\) of the sums is especially important from this viewpoint. In this connection, the problem arises of finding a quantity \(A_n\) (as simple as possible) such that for some constant \(C>0\) one has \(C^{-1} A_n \leq \| S_n \|_p \leq C A_n\). For the system of Rademacher functions, this leads to the classical Khintchine inequalities from 1923 [A. Khintchine, Math. Z. 18, 289–306 (1923; JFM 49.0159.03)]. For general systems of independent functions, Rosenthal in 1970 proved a far-reaching generalization [H. P. Rosenthal, Isr. J. Math. 8, 273-303 (1970; Zbl 0213.19303)].
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. This 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.
The survey spans over more than 70 pages, with a bibliography of 87 titles, many of which are taken from the Russian literature – but many thanks to the authors for providing the reader with the exact references to their English translations. Moreover, the authors provide many proofs or sketches thereof, so that the interested reader gets a fairly good idea about the used techniques with no need to look up the vast literature immediately.

Reviewer: Carsten Michels (Oldenburg)

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46-02	Research exposition (monographs, survey articles) pertaining to functional analysis
46B09	Probabilistic methods in Banach space theory
46B20	Geometry and structure of normed linear spaces
60B11	Probability theory on linear topological spaces
46B70	Interpolation between normed linear spaces

Keywords:

independent functions; Khinchin inequalities; Kruglov property; Rosenthal inequalities; Kruglov operator; symmetric space; Orlicz space; Marcinkiewicz space; Lorentz space; Boyd indices; \(K\)-functional; real method of interpolation; integral-uniform norm

Citations:

JFM 49.0159.03; Zbl 0213.19303