### 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 L_{1} 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 language | English |
---|---|

Pages (from-to) | 17-23 |

Number of pages | 7 |

Journal | Information Processing Letters |

Volume | 51 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 12 1994 |

Externally published | Yes |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Information Processing Letters*, vol. 51, no. 1, pp. 17-23. https://doi.org/10.1016/0020-0190(94)00061-1

}

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 -