On the relationship between ε-biased random variables and ε-dependent random variables

Shuji Jinbo, Akira Maruoka

Research output: Contribution to journalArticle

Abstract

The notions of "k-wise ε-dependent" and "k-wise ε-biased" are somewhat weaker in randomness than those of independent random variables. Random variables with these properties could be substitutes for independent random variables of randomized algorithms. In this paper, after giving relevant definitions for these notions, the relationship between these notions is presented: For any integer k and n with 1 ≤ k ≤ n, if a system of n random variables is k-wise ε-biased, then it is k-wise 4(1 - 2-k)ε-dependent in maximum norm and k-wise 2(1 - 2-k)ε-dependent in L1 norm with respect to the uniform distribution. It has been presented, in literature, that k-wise ε-biased random variables are substituted for k-wise δ-dependent random variables in many randomized algorithms, so the results of this paper are expected to reduce the running time of resultant algorithms due to derandomization.

Original languageEnglish
Pages (from-to)17-23
Number of pages7
JournalInformation Processing Letters
Volume51
Issue number1
DOIs
Publication statusPublished - Jul 12 1994
Externally publishedYes

Fingerprint

Dependent Random Variables
Random variables
Biased
Random variable
Independent Random Variables
Randomized Algorithms
Dependent
Derandomization
Maximum Norm
L1-norm
Substitute
Uniform distribution
Randomness
Integer
Relationships

Keywords

  • Algorithms
  • Distribution
  • Independence
  • Linear algebra
  • Random variables
  • Randomized algorithms
  • Sample space

ASJC Scopus subject areas

  • Computational Theory and Mathematics

Cite this

On the relationship between ε-biased random variables and ε-dependent random variables. / Jinbo, Shuji; Maruoka, Akira.

In: Information Processing Letters, Vol. 51, No. 1, 12.07.1994, p. 17-23.

Research output: Contribution to journalArticle

@article{f1d37f6e77e4453287949ba50e3c68eb,
title = "On the relationship between ε-biased random variables and ε-dependent random variables",
abstract = "The notions of {"}k-wise ε-dependent{"} and {"}k-wise ε-biased{"} are somewhat weaker in randomness than those of independent random variables. Random variables with these properties could be substitutes for independent random variables of randomized algorithms. In this paper, after giving relevant definitions for these notions, the relationship between these notions is presented: For any integer k and n with 1 ≤ k ≤ n, if a system of n random variables is k-wise ε-biased, then it is k-wise 4(1 - 2-k)ε-dependent in maximum norm and k-wise 2(1 - 2-k)ε-dependent in L1 norm with respect to the uniform distribution. It has been presented, in literature, that k-wise ε-biased random variables are substituted for k-wise δ-dependent random variables in many randomized algorithms, so the results of this paper are expected to reduce the running time of resultant algorithms due to derandomization.",
keywords = "Algorithms, Distribution, Independence, Linear algebra, Random variables, Randomized algorithms, Sample space",
author = "Shuji Jinbo and Akira Maruoka",
year = "1994",
month = "7",
day = "12",
doi = "10.1016/0020-0190(94)00061-1",
language = "English",
volume = "51",
pages = "17--23",
journal = "Information Processing Letters",
issn = "0020-0190",
publisher = "Elsevier",
number = "1",

}

TY - JOUR

T1 - On the relationship between ε-biased random variables and ε-dependent random variables

AU - Jinbo, Shuji

AU - Maruoka, Akira

PY - 1994/7/12

Y1 - 1994/7/12

N2 - The notions of "k-wise ε-dependent" and "k-wise ε-biased" are somewhat weaker in randomness than those of independent random variables. Random variables with these properties could be substitutes for independent random variables of randomized algorithms. In this paper, after giving relevant definitions for these notions, the relationship between these notions is presented: For any integer k and n with 1 ≤ k ≤ n, if a system of n random variables is k-wise ε-biased, then it is k-wise 4(1 - 2-k)ε-dependent in maximum norm and k-wise 2(1 - 2-k)ε-dependent in L1 norm with respect to the uniform distribution. It has been presented, in literature, that k-wise ε-biased random variables are substituted for k-wise δ-dependent random variables in many randomized algorithms, so the results of this paper are expected to reduce the running time of resultant algorithms due to derandomization.

AB - The notions of "k-wise ε-dependent" and "k-wise ε-biased" are somewhat weaker in randomness than those of independent random variables. Random variables with these properties could be substitutes for independent random variables of randomized algorithms. In this paper, after giving relevant definitions for these notions, the relationship between these notions is presented: For any integer k and n with 1 ≤ k ≤ n, if a system of n random variables is k-wise ε-biased, then it is k-wise 4(1 - 2-k)ε-dependent in maximum norm and k-wise 2(1 - 2-k)ε-dependent in L1 norm with respect to the uniform distribution. It has been presented, in literature, that k-wise ε-biased random variables are substituted for k-wise δ-dependent random variables in many randomized algorithms, so the results of this paper are expected to reduce the running time of resultant algorithms due to derandomization.

KW - Algorithms

KW - Distribution

KW - Independence

KW - Linear algebra

KW - Random variables

KW - Randomized algorithms

KW - Sample space

UR - http://www.scopus.com/inward/record.url?scp=0028466678&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028466678&partnerID=8YFLogxK

U2 - 10.1016/0020-0190(94)00061-1

DO - 10.1016/0020-0190(94)00061-1

M3 - Article

AN - SCOPUS:0028466678

VL - 51

SP - 17

EP - 23

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 1

ER -