# Gebelein's inequality and its consequences

Banach Center Publications (2006)

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

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topM. 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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