Gebelein's inequality and its consequences

M. Beśka, and Z. Ciesielski. "Gebelein's inequality and its consequences." Banach Center Publications 72.1 (2006): 11-23. <http://eudml.org/doc/282570>.

@article{M2006,
abstract = {Let $(X_i, i=1,2,...)$ be the normalized gaussian system such that $X_i ∈ N(0,1)$, i = 1,2,... and let the correlation matrix $ρ_\{ij\} = E(X_iX_j)$ satisfy the following hypothesis: $C = sup_\{i≥1\} ∑_\{j=1\}^\{∞\} |ρ_\{i,j\}| < ∞$. 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 ∈ L¹(ν) with (f,1)ν = 0.},
author = {M. Beśka, Z. Ciesielski},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {11-23},
title = {Gebelein's inequality and its consequences},
url = {http://eudml.org/doc/282570},
volume = {72},
year = {2006},
}

TY - JOUR
AU - M. Beśka
AU - Z. Ciesielski
TI - Gebelein's inequality and its consequences
JO - Banach Center Publications
PY - 2006
VL - 72
IS - 1
SP - 11
EP - 23
AB - Let $(X_i, i=1,2,...)$ be the normalized gaussian system such that $X_i ∈ N(0,1)$, i = 1,2,... and let the correlation matrix $ρ_{ij} = E(X_iX_j)$ satisfy the following hypothesis: $C = sup_{i≥1} ∑_{j=1}^{∞} |ρ_{i,j}| < ∞$. 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 ∈ L¹(ν) with (f,1)ν = 0.
LA - eng
UR - http://eudml.org/doc/282570
ER -