Gebelein's inequality and its consequences

Abstract

Let (Xi, i = 1, 2,. . .) be the normalized gaussian system such that Xi ∈ N (0, 1), i = 1, 2,. .. and let the correlation matrix ρij = E(XiXj) satisfy the following hypothesis: C = sup i≥1 ∞ j=1 |ρi,j| < ∞. We present Gebelein's inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy's norm for the gaussian sequence etc. The main result is that f (X1) + • • • + f (Xn) n → 0 a.s. for f ∈ L 1 (ν) with (f, 1)ν = 0.

Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion

“…The presented results generalize several results from [1,2], where the case that X = Y = R is studied.…”

“…With the same argument as in [2] one can obtain the following extension of [2, Lemma 2.1] to the vector valued setting. …”

“…Let the correlation number between ξ and η be defined as ρ = |Eξη|. The Gebelein inequality [13] (also see [2]) states that for all f, g : R → R such that f (ξ ), g(η) ∈ L 2 ( ) and E f (ξ ) = g(η) = 0. In other words, if Eq.…”

“…In [1,2], Beśka and Ciesielski have studied sequences of real valued centered Gaussian random variables (ξ i ) n≥1 with Eξ 2 i = 1 and (ξ 1 , ξ 2 , . .…”

Besov-Orlicz path regularity of non-Gaussian processes

“…In order to establish such Besov-Orlicz regularity results, one would hope to proceed as in [24] (or [4]). The proofs there rely on Gebelein's inequality [9] (see also [3]):…”