Rademacher chaos: tail estimates versus limit theorems

Abstract

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.

Rademacher Chaos in Symmetric Spaces

“…In contrast to that, as we will see below, asymptotically the sums S N have the standard normal distribution, and thus the functions r  ,  ∈ A, are asymptotically independent like the usual Rademacher functions. This "divergence" in estimates for the moments of a Rademacher fractional chaos and its asymptotic behaviour was previously observed in the paper [14]. To justify the last assertion, we will use Theorem 1.7 from [14].…”

“…This "divergence" in estimates for the moments of a Rademacher fractional chaos and its asymptotic behaviour was previously observed in the paper [14]. To justify the last assertion, we will use Theorem 1.7 from [14].…”

“…In the concluding part of the paper, we show that every uniformly bounded Bessel system (in particular, any Rademacher chaos) in a symmetric space X, satisfying the embedding ExpL 2 ⊂ X, has the random unconditional convergence property, which is in a certain sense opposite to the RUD property. In addition, we give here a concrete example illustrating the interesting fact of "divergence" of the moment estimates of a Rademacher fractional chaos and its asymptotic behaviour (see also [14]).…”

“…[2, Proposition 2.2]). The next step in the study of the behaviour of the Rademacher chaos in symmetric spaces was made by the authors of this paper by using the important concept of combinatorial dimension developed earlier by Blei (see [10,11,12,13,14]) . Namely, in [9], it was shown that the above results in [7] and [8] can be extended to a non-complete chaos…”

On unconditionality of fractional Rademacher chaos in symmetric spaces

“…Note that the Rademacher system itself is an unconditional (and even symmetric with constant 1) basic sequence in any symmetric space (see for example, Proposition 2.2 in [2]). The next step in the study of the behaviour of the Rademacher chaos in symmetric spaces was made by the authors of the present paper by employing the important concept of combinatorial dimension developed earlier by Blei (see [10]- [14]). Namely, in [9] it was shown that the above results in [7] and [8] can be extended to a non-complete chaos…”