S. V. Astashkin

Series of independent random variables in rearrangement invariant spaces: An operator approach

This paper studies series of independent random variables in rearrangement invariant spacesX on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(Lp), 1≤p≤∞ and Lorentz spacesAΨ. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumptionLp ⊂X, p<∞ is sufficient to guarantee that convergence of such series inX (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence inX of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result.

Disjointly Strictly Singular Inclusions of Symmetric Spaces

In this paper, the disjoint strict singularity of inclusions of symmetric spaces of functions on an interval is considered. A condition for the presence of a “gap” between spaces sufficient for the inclusion of one of these spaces into the other to be disjointly strictly singular is found. The condition is stated in terms of fundamental functions of spaces and is exact in a certain sense. In parallel, necessary and sufficient conditions for an inclusion of Lorentz spaces to be disjointly strictly singular (and similar conditions for Marcinkiewicz spaces) are obtained and certain other assertions are proved.

Best constants in Rosenthal-type inequalities and the Kruglov operator

Let X be a symmetric Banach function space on [0, 1] with the Kruglov property, and let f = {f k } n k=1 , n ≥ 1 be an arbitrary sequence of independent random variables in X. This paper presents sharp estimates in the deterministic characterization of the quantities.

Extraction of Subsystems "Majorized" by the Rademacher System

AbstractIn this paper it is proved that from any uniformly bounded orthonormal system {fn}n=1∞of random variables defined on the probability space (Ω, ε, P), one can extract a subsystem {fni}iEmphasis>=1/∞majorized in distribution by the Rademacher system on [0, 1]. This means that {

A weighted Khintchine inequality.

P\left\{ {\omega \in \Omega :\left| {\sum\limits_{i = 1}^m {a_i f_{n_i } } \left( \omega \right)} \right| > z} \right\} \leqslant C\left| {\left\{ {t \in \left[ {0,1} \right]:\left| {\sum\limits_{i = 1}^m {a_i r_i \left( t \right)} } \right| > \frac{z}{C}} \right\}} \right|.

Lieb-Thirring inequality for L p norms

}, whereC>0 is independent of m∈N, ai∈N (i=1,…,m) andz>0.

Some New Extrapolation Estimates for the Scale of Spaces

Abstract We show that the multiplicator space M (X) of an rearrangement invariant (r.i.) space X on [0,1] and the nice part N 0 ( X ) of X , that is, the set of all a ∈ X for which the subspaces generated by sequences of dilations and translations of a are uniformly complemented, coincide when the space X is separable. In the general case, the nice part is larger than the multiplicator space. Several examples of descriptions of M (X) and N 0 ( X ) for concrete X are presented.

A probabilistic version of Rosenthal's inequality

i=1 ai 1/2 , for every (ai) ∈ l, where (ri) are the Rademacher functions, that is, ri(t) := sign sin(2πt), t ∈ [0, 1], i ∈ N. A weighted version of the above inequality was recently proved in [18]. Namely, let w be a weight satisfying the following conditions (a) for some q > p we have w ∈ L([0, 1]); (b) the support of w satisfies m(supp(w)) > 2/3. Then there exist constants C1, C2 > 0, depending on p and w, such that for every a = (ai) ∈ l