# On families of weakly dependent random variables

• Volume: 95, Issue: 1, page 123-132
• ISSN: 0137-6934

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## Abstract

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Let ${ₙ}^{\left(k\right)}$ be a family of random independent k-element subsets of [n] = 1,2,...,n and let $\left({ₙ}^{\left(k\right)},\ell \right)={ₙ}^{\left(k\right)}\left(\ell \right)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to ${ₙ}^{\left(k\right)}\left(\ell \right)$ depends only on the edges of ${ₙ}^{\left(k\right)}$ contained in S. Then, the edges of ${ₙ}^{\left(k\right)}\left(\ell \right)$ are ’weakly dependent’, say, the events that two given subsets S and T are in ${ₙ}^{\left(k\right)}\left(\ell \right)$ are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We also list some questions which, despite the progress which has been made for the last few years, remain to puzzle researchers who work in the area of probabilistic combinatorics.

## How to cite

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Tomasz Łuczak. "On families of weakly dependent random variables." Banach Center Publications 95.1 (2011): 123-132. <http://eudml.org/doc/282446>.

@article{TomaszŁuczak2011,
abstract = {Let $ₙ^\{(k)\}$ be a family of random independent k-element subsets of [n] = 1,2,...,n and let $(ₙ^\{(k)\},ℓ) = ₙ^\{(k)\}(ℓ)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to $ₙ^\{(k)\}(ℓ)$ depends only on the edges of $ₙ^\{(k)\}$ contained in S. Then, the edges of $ₙ^\{(k)\}(ℓ)$ are ’weakly dependent’, say, the events that two given subsets S and T are in $ₙ^\{(k)\}(ℓ)$ are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We also list some questions which, despite the progress which has been made for the last few years, remain to puzzle researchers who work in the area of probabilistic combinatorics.},
author = {Tomasz Łuczak},
journal = {Banach Center Publications},
keywords = {random graph; hypergraph; arithmetic progression; limit theorem; extremal properties; large deviation; dependence},
language = {eng},
number = {1},
pages = {123-132},
title = {On families of weakly dependent random variables},
url = {http://eudml.org/doc/282446},
volume = {95},
year = {2011},
}

TY - JOUR
AU - Tomasz Łuczak
TI - On families of weakly dependent random variables
JO - Banach Center Publications
PY - 2011
VL - 95
IS - 1
SP - 123
EP - 132
AB - Let $ₙ^{(k)}$ be a family of random independent k-element subsets of [n] = 1,2,...,n and let $(ₙ^{(k)},ℓ) = ₙ^{(k)}(ℓ)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to $ₙ^{(k)}(ℓ)$ depends only on the edges of $ₙ^{(k)}$ contained in S. Then, the edges of $ₙ^{(k)}(ℓ)$ are ’weakly dependent’, say, the events that two given subsets S and T are in $ₙ^{(k)}(ℓ)$ are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We also list some questions which, despite the progress which has been made for the last few years, remain to puzzle researchers who work in the area of probabilistic combinatorics.
LA - eng
KW - random graph; hypergraph; arithmetic progression; limit theorem; extremal properties; large deviation; dependence
UR - http://eudml.org/doc/282446
ER -

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