Gebelein's inequality and its consequences

M. Beśka; Z. Ciesielski

Banach Center Publications (2006)

  • Volume: 72, Issue: 1, page 11-23
  • ISSN: 0137-6934

Abstract

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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 i j = E ( X i X j ) satisfy the following hypothesis: C = s u p 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.

How to cite

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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 -

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