# On families of weakly dependent random variables

Banach Center Publications (2011)

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

## Access Full Article

top## Abstract

top## How to cite

topTomasz 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.