Rademacher chaos: tail estimates versus limit theorems

Abstract

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.

Stein's Method and Stochastic Analysis of Rademacher Functionals

“…In particular, in [21] a crucial role is played by kernels "with low influences": we will point out in Section 4 that influence indices are an important component of our bounds for the normal approximation of elements of a fixed chaos. In Section 6.3, we also show that the combination of the findings of the present paper with those of [21] yields an elegant answer to a question left open by Blei and Janson in [4].…”

“…IV]) and Rademacher chaos (see e.g. [4,11,18]). In Meyer's monograph [20] the collection {J q (f ) : f ∈ ℓ 2 0 (N)…”

“…In the paper [4], Blei and Janson studied the problem of finding conditions on the set F N , in order to have that the CLT In order to state Blei and Janson's result, we need to introduce some more notation.…”

“…Gaussian sequence G = {G i : i 1}, where each G i has mean zero and unit variance. A natural question, which has been left open by Blei and Janson in [4,Remark 4.6], is whether under the conditions (6.57)-(6.58) the sequence S G N , N 1, converges in law towards a standard Gaussian distribution. Note that this problem could be tackled by a direct computation, based for instance on [22] or [27].…”

“…One striking application of these techniques is given in Section 6, where we deduce explicit bounds (as well as a new proof) for a combinatorial central limit theorem (CLT) recently proved by Blei and Janson in [4], involving sequences of multiple integrals over finite "sparse" sets. In the original Blei and Janson's proof, the required sparseness condition emerges as an efficient way of re-expressing moment computations related to martingale CLTs.…”