## Abstract

We study the discrepancy of jittered sampling sets: such a set P⊂ [0,1]^{d} is generated for fixed m∈ℕ by partitioning [0,1]^{d} into m^{d} axis aligned cubes of equal measure and placing a random point inside each of the N=m^{d} cubes. We prove that, for N sufficiently large, 1/10 d/N^{1/2+1/2d} ≤ED_{N}∗(P)≤ √d(log N) 1/2/N^{1/2+1/2d}, where the upper bound with an unspecified constant C_{d} was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N≳d^{d}. We also prove a partition principle showing that every partition of [0,1]^{d} combined with a jittered sampling construction gives rise to a set whose expected squared L^{2}-discrepancy is smaller than that of purely random points.

Original language | English |
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Pages (from-to) | 199-216 |

Number of pages | 18 |

Journal | Journal of Complexity |

Volume | 33 |

Early online date | 17 Nov 2015 |

DOIs | |

Publication status | Published - Apr 2016 |

## Keywords

- Inverse of the star discrepancy
- Jittered sampling
- L-discrepancy
- Star discrepancy

## ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Control and Optimization
- Applied Mathematics